# Graph Theory

This section of Discrete Mathematics & Theoretical Computer Science seeks high quality articles on structural and algorithmic aspects of graphs and related discrete mathematical models. We particularly seek topics with an intersection between discrete mathematics and computer science. We handle submissions in all areas of finite graph theory.

### Even cycles and perfect matchings in claw-free plane graphs

Lov{á}sz showed that a matching covered graph $G$ has an ear decomposition starting with an arbitrary edge of $G$. Let $G$ be a graph which has a perfect matching. We call $G$ cycle-nice if for each even cycle $C$ of $G$, $G-V(C)$ has a perfect matching. If $G$ is a cycle-nice matching covered graph, then $G$ has ear decompositions starting with an arbitrary even cycle of $G$. In this paper, we characterize cycle-nice claw-free plane graphs. We show that the only cycle-nice simple 3-connected claw-free plane graphs are $K_4$, $W_5$ and $\overline C_6$. Furthermore, every cycle-nice 2-connected claw-free plane graph can be obtained from a graph in the family ${\cal F}$ by a sequence of three types of operations, where ${\cal F}$ consists of even cycles, a diamond, $K_4$, and $\overline C_6$.

### (Open) packing number of some graph products

The packing number of a graph $G$ is the maximum number of closed neighborhoods of vertices in $G$ with pairwise empty intersections. Similarly, the open packing number of $G$ is the maximum number of open neighborhoods in $G$ with pairwise empty intersections. We consider the packing and open packing numbers on graph products. In particular we give a complete solution with respect to some properties of factors in the case of lexicographic and rooted products. For Cartesian, strong and direct products, we present several lower and upper bounds on these parameters.

### The agreement distance of unrooted phylogenetic networks

A rearrangement operation makes a small graph-theoretical change to a phylogenetic network to transform it into another one. For unrooted phylogenetic trees and networks, popular rearrangement operations are tree bisection and reconnection (TBR) and prune and regraft (PR) (called subtree prune and regraft (SPR) on trees). Each of these operations induces a metric on the sets of phylogenetic trees and networks. The TBR-distance between two unrooted phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest, that is, a forest with a minimum number of components that covers both $T$ and $T'$ in a certain way. This characterisation has facilitated the development of fixed-parameter tractable algorithms and approximation algorithms. Here, we introduce maximum agreement graphs as a generalisations of maximum agreement forests for phylogenetic networks. While the agreement distance -- the metric induced by maximum agreement graphs -- does not characterise the TBR-distance of two networks, we show that it still provides constant-factor bounds on the TBR-distance. We find similar results for PR in terms of maximum endpoint agreement graphs.

### The Complexity of Helly-$B_{1}$ EPG Graph Recognition

Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the intersection graph class of edge paths on a grid. An EPG graph $G$ is a graph that admits a representation where its vertices correspond to paths in a grid $Q$, such that two vertices of $G$ are adjacent if and only if their corresponding paths in $Q$ have a common edge. If the paths in the representation have at most $k$ bends, we say that it is a $B_k$-EPG representation. A collection $C$ of sets satisfies the Helly property when every sub-collection of $C$ that is pairwise intersecting has at least one common element. In this paper, we show that given a graph $G$ and an integer $k$, the problem of determining whether $G$ admits a $B_k$-EPG representation whose edge-intersections of paths satisfy the Helly property, so-called Helly-$B_k$-EPG representation, is in NP, for every $k$ bounded by a polynomial function of $|V(G)|$. Moreover, we show that the problem of recognizing Helly-$B_1$-EPG graphs is NP-complete, and it remains NP-complete even when restricted to 2-apex and 3-degenerate graphs.

### Antifactors of regular bipartite graphs

Let $G=(X,Y;E)$ be a bipartite graph, where $X$ and $Y$ are color classes and $E$ is the set of edges of $G$. Lovász and Plummer \cite{LoPl86} asked whether one can decide in polynomial time that a given bipartite graph $G=(X,Y; E)$ admits a 1-anti-factor, that is subset $F$ of $E$ such that $d_F(v)=1$ for all $v\in X$ and $d_F(v)\neq 1$ for all $v\in Y$. Cornuéjols \cite{CHP} answered this question in the affirmative. Yu and Liu \cite{YL09} asked whether, for a given integer $k\geq 3$, every $k$-regular bipartite graph contains a 1-anti-factor. This paper answers this question in the affirmative.

### The super-connectivity of Johnson graphs

For positive integers $n,k$ and $t$, the uniform subset graph $G(n, k, t)$ has all $k$-subsets of $\{1,2,\ldots, n\}$ as vertices and two $k$-subsets are joined by an edge if they intersect at exactly $t$ elements. The Johnson graph $J(n,k)$ corresponds to $G(n,k,k-1)$, that is, two vertices of $J(n,k)$ are adjacent if the intersection of the corresponding $k$-subsets has size $k-1$. A super vertex-cut of a connected graph is a set of vertices whose removal disconnects the graph without isolating a vertex and the super-connectivity is the size of a minimum super vertex-cut. In this work, we fully determine the super-connectivity of the family of Johnson graphs $J(n,k)$ for $n\geq k\geq 1$.

### The Chromatic Number of the Disjointness Graph of the Double Chain

Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different colors. In this paper we show that if $P$ is the point configuration known as the double chain, with $k$ points in the upper convex chain and $l \ge k$ points in the lower convex chain, then $k+l- \left\lfloor \sqrt{2l+\frac{1}{4}} - \frac{1}{2}\right\rfloor$ colors are needed and that this number is sufficient.

### A method for eternally dominating strong grids

In the eternal domination game, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices and no more than one guard may occupy a vertex. The goal is to determine the eternal domination number of a graph which is the minimum number of guards required to defend the graph against an infinite sequence of attacks. In this paper, we continue the study of the eternal domination game on strong grids. Cartesian grids have been vastly studied with tight bounds for small grids such as 2×n, 3×n, 4×n, and 5×n grids, and recently it was proven in [Lamprou et al., CIAC 2017, 393-404] that the eternal domination number of these grids in general is within O(m + n) of their domination number which lower bounds the eternal domination number. Recently, Finbow et al. proved that the eternal domination number of strong grids is upper bounded by mn 6 + O(m + n). We adapt the techniques of [Lamprou et al., CIAC 2017, 393-404] to prove that the eternal domination number of strong grids is upper bounded by mn 7 + O(m + n). While this does not improve upon a recently announced bound of ⎡m/3⎤ x⎡n/3⎤ + O(m √ n) [Mc Inerney, Nisse, Pérennes, HAL archives, 2018; Mc Inerney, Nisse, Pérennes, CIAC 2019] in the general case, we show that our bound is an improvement in the case where the smaller of the two dimensions is at most 6179.

### On the Complexity of Digraph Colourings and Vertex Arboricity

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.

### From light edges to strong edge-colouring of 1-planar graphs

A strong edge-colouring of an undirected graph $G$ is an edge-colouring where every two edges at distance at most~$2$ receive distinct colours. The strong chromatic index of $G$ is the least number of colours in a strong edge-colouring of $G$. A conjecture of Erdős and Nešet\v{r}il, stated back in the $80$'s, asserts that every graph with maximum degree $\Delta$ should have strong chromatic index at most roughly $1.25 \Delta^2$. Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly $4\Delta$, and even to smaller values under additional structural requirements.In this work, we initiate the study of the strong chromatic index of $1$-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of $1$-planar graphs with maximum degree~$\Delta$ and strong chromatic index roughly $6\Delta$. As an upper bound, we prove that the strong chromatic index of a $1$-planar graph with maximum degree $\Delta$ is at most roughly $24\Delta$ (thus linear in $\Delta$). The proof of this result is based on the existence of light edges in $1$-planar graphs with minimum degree at least~$3$.

### Power domination in maximal planar graphs

Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. Then iteratively, whenever some monitored vertex v has a single neighbor u not yet monitored, u gets monitored. A set S is said to be a power dominating set of the graph G if all vertices of G eventually are monitored. The power domination number of a graph is the minimum size of a power dominating set. In this paper, we prove that any maximal planar graph of order n ≥ 6 admits a power dominating set of size at most (n−2)/4 .

### Generalized Petersen graphs and Kronecker covers

The family of generalized Petersen graphs $G(n,k)$, introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover $KC(G)$ of a simple undirected graph $G$ is a a special type of bipartite covering graph of $G$, isomorphic to the direct (tensor) product of $G$ and $K_2$. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe that some of such quotients are again generalized Petersen graphs, and describe all such pairs.The results of this paper have been presented at EUROCOMB 2019 and an extended abstract has been published elsewhere.

### Equitable Coloring and Equitable Choosability of Planar Graphs without chordal 4- and 6-Cycles

A graph $G$ is equitably $k$-choosable if, for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices. A graph is equitably $k$-colorable if the vertex set $V(G)$ can be partitioned into $k$ independent subsets $V_1$, $V_2$, $\cdots$, $V_k$ such that $||V_i|-|V_j||\leq 1$ for $1\leq i, j\leq k$. In this paper, we prove that if $G$ is a planar graph without chordal $4$- and $6$-cycles, then $G$ is equitably $k$-colorable and equitably $k$-choosable where $k\geq\max\{\Delta(G), 7\}$.

### Monochromatic loose paths in multicolored $k$-uniform cliques

For integers $k\ge 2$ and $\ell\ge 0$, a $k$-uniform hypergraph is called a loose path of length $\ell$, and denoted by $P_\ell^{(k)}$, if it consists of $\ell$ edges $e_1,\dots,e_\ell$ such that $|e_i\cap e_j|=1$ if $|i-j|=1$ and $e_i\cap e_j=\emptyset$ if $|i-j|\ge2$. In other words, each pair of consecutive edges intersects on a single vertex, while all other pairs are disjoint. Let $R(P_\ell^{(k)};r)$ be the minimum integer $n$ such that every $r$-edge-coloring of the complete $k$-uniform hypergraph $K_n^{(k)}$ yields a monochromatic copy of $P_\ell^{(k)}$. In this paper we are mostly interested in constructive upper bounds on $R(P_\ell^{(k)};r)$, meaning that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of $P_\ell^{(k)}$ in every coloring. In particular, we show that there is a constant $c>0$ such that for all $k\ge 2$, $\ell\ge3$, $2\le r\le k-1$, and $n\ge k(\ell+1)r(1+\ln(r))$, there is an algorithm such that for every $r$-edge-coloring of the edges of $K_n^{(k)}$, it finds a monochromatic copy of $P_\ell^{(k)}$ in time at most $cn^k$. We also prove a non-constructive upper bound $R(P_\ell^{(k)};r)\le(k-1)\ell r$.

### Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.

### Constrained ear decompositions in graphs and digraphs

Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously N P-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovász states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In this paper, we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most is polynomial-time solvable for all fixed positive integer. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in F is N P-complete for any finite set F of positive integers. We also prove that, for any k ≥ 2, deciding whether a graph admits an ear decomposition with all ears of length 0 mod k is N P-complete. We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most is polynomial-time solvable for all positive integer ; deciding whether a digraph admits a handle decomposition without handles of length in F is N P-complete for […]

### The irregularity of two types of trees

The irregularity of a graph $G$ is defined as the sum of weights $|d(u)-d(v)|$ of all edges $uv$ of $G$, where $d(u)$ and $d(v)$ are the degrees of the vertices $u$ and $v$ in $G$, respectively. In this paper, some structural properties on trees with maximum (or minimum) irregularity among trees with given degree sequence and trees with given branching number are explored, respectively. Moreover, the corresponding trees with maximum (or minimum) irregularity are also found, respectively.

### Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph

A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open domination graph when $H$ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of $V(G)$. For the class of trees when $H$ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products $G \Box H$ when $H$ is a 5-cycle or a 4-cycle.

### Planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable

For planar graphs, we consider the problems of list edge coloring and list total coloring. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list $\Delta$-edge-colorable or list $(\Delta +1)$-total-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta \geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta \geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k (\forall 3 \leq k \leq 6)$ are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.

### Rainbow eulerian multidigraphs and the product of cycles

An arc colored eulerian multidigraph with $l$ colors is rainbow eulerian if there is an eulerian circuit in which a sequence of $l$ colors repeats. The digraph product that refers the title was introduced by Figueroa-Centeno et al. as follows: let $D$ be a digraph and let $\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\in \Gamma$. Consider any function $h:E(D) \longrightarrow \Gamma$. Then the product $D \otimes_h \Gamma$ is the digraph with vertex set $V(D) \times V$ and $((a,x),(b,y)) \in E(D \otimes_h \Gamma)$ if and only if $(a,b) \in E(D)$ and $(x,y) \in E(h (a,b))$. In this paper we use rainbow eulerian multidigraphs and permutations as a way to characterize the $\otimes_h$-product of oriented cycles. We study the behavior of the $\otimes_h$-product when applied to digraphs with unicyclic components. The results obtained allow us to get edge-magic labelings of graphs formed by the union of unicyclic components and with different magic sums.

### Heredity for generalized power domination

In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma_{p,k}(G-e)$, $\gamma_{p,k}(G/e)$ and for $\gamma_{p,k}(G-v)$ in terms of $\gamma_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $\gamma_{p,k}(G-e) = \gamma_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.

### Open k-monopolies in graphs: complexity and related concepts

Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values.

### Edge-partitioning graphs into regular and locally irregular components

A graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graph $G$, there exists a minimum integer $\chi^{\prime}_{\mathrm{irr}}(G)$ such that $G$ admits an edge-partition into $\chi^{\prime}_{\mathrm{irr}}(G)$ classes, each of which induces a locally irregular graph. In particular, they conjectured that $\chi^{\prime}_{\mathrm{irr}}(G) \leq 3$ for every $G$, unless $G$ belongs to a well-characterized family of non-decomposable graphs. This conjecture is far from being settled, as notably (1) no constant upper bound on$\chi^{\prime}_{\mathrm{irr}}(G)$ is known for $G$ bipartite, and (2) no satisfactory general upper bound on $\chi^{\prime}_{\mathrm{irr}}(G)$ is known. We herein investigate the consequences on this question of allowing a decomposition to include regular components as well. As a main result, we prove that every bipartite graph admits such a decomposition into at most $6$ subgraphs. This result implies that every graph $G$ admits a decomposition into at most $6(\lfloor \mathrm{log} \chi (G) \rfloor +1)$ subgraphs whose components are regular or locally irregular.

### The complexity of deciding whether a graph admits an orientation with fixed weak diameter

An oriented graph $\overrightarrow{G}$ is said weak (resp. strong) if, for every pair $\{ u,v \}$ of vertices of $\overrightarrow{G}$, there are directed paths joining $u$ and $v$ in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most $k$, we call $\overrightarrow{G}$ $k$-weak (resp. $k$-strong). We consider several problems asking whether an undirected graph $G$ admits orientations satisfying some connectivity and distance properties. As a main result, we show that deciding whether $G$ admits a $k$-weak orientation is NP-complete for every $k \geq 2$. This notably implies the NP-completeness of several problems asking whether $G$ is an extremal graph (in terms of needed colours) for some vertex-colouring problems.

### Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights

Let $G$ be a graph and $\mathcal{S}$ be a subset of $Z$. A vertex-coloring $\mathcal{S}$-edge-weighting of $G$ is an assignment of weights by the elements of $\mathcal{S}$ to each edge of $G$ so that adjacent vertices have different sums of incident edges weights. It was proved that every 3-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} = \{1,2 \}$ (H. Lu, Q. Yu and C. Zhang, Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32 (2011), 22-27). In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$. These bounds we obtain are tight, since there exists a family of infinite bipartite graphs which are 2-connected and do not admit vertex-coloring $\mathcal{S}$-edge-weightings for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$.

### The double competition multigraph of a digraph

In this article, we introduce the notion of the double competition multigraph of a digraph. We give characterizations of the double competition multigraphs of arbitrary digraphs, loopless digraphs, reflexive digraphs, and acyclic digraphs in terms of edge clique partitions of the multigraphs.

### The game colouring number of powers of forests

We prove that the game colouring number of the $m$-th power of a forest of maximum degree $\Delta\ge3$ is bounded from above by $\frac{(\Delta-1)^m-1}{\Delta-2}+2^m+1,$ which improves the best known bound by an asymptotic factor of 2.

### Cubical coloring — fractional covering by cuts and semidefinite programming

We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

### Disimplicial arcs, transitive vertices, and disimplicial eliminations

In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A diclique of a digraph is a pair $V$ → $W$ of sets of vertices such that $v$ → $w$ is an arc for every $v$ ∈ $V$ and $w$ ∈ $W$. An arc $v$ → $w$ is disimplicial when it belongs to a unique maximal diclique. We show that the problem of finding the disimplicial arcs is equivalent, in terms of time and space complexity, to that of locating the transitive vertices. As a result, an efficient algorithm to find the bisimplicial edges of bipartite graphs is obtained. Then, we develop simple algorithms to build disimplicial elimination schemes, which can be used to generate bisimplicial elimination schemes for bipartite graphs. Finally, we study two classes related to perfect disimplicial elimination digraphs, namely weakly diclique irreducible digraphs and diclique irreducible digraphs. The former class is associated to finite posets, while the latter corresponds to dedekind complete finite posets.

### Packing Plane Perfect Matchings into a Point Set

Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? For points in general position we prove the lower bound of ⌊log2$n$⌋$-1$. For some special configurations of point sets, we give the exact answer. We also consider some restricted variants of this problem.

### The complexity of $P$4-decomposition of regular graphs and multigraphs

Let G denote a multigraph with edge set E(G), let µ(G) denote the maximum edge multiplicity in G, and let Pk denote the path on k vertices. Heinrich et al.(1999) showed that P4 decomposes a connected 4-regular graph G if and only if |E(G)| is divisible by 3. We show that P4 decomposes a connected 4-regular multigraph G with µ(G) ≤2 if and only if no 3 vertices of G induce more than 4 edges and |E(G)| is divisible by 3. Oksimets (2003) proved that for all integers k ≥3, P4 decomposes a connected 2k-regular graph G if and only if |E(G)| is divisible by 3. We prove that for all integers k ≥2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete. El-Zanati et al.(2014) showed that for all integers k ≥1, every 6k-regular multigraph with µ(G) ≤2k has a P4-decomposition. We show that unless P = NP, this result is best possible with respect to µ(G) by proving that for all integers k ≥3 the problem of determining if P4 decomposes a 2k-regular multigraph with µ(G) ≤⌊2k / 3 ⌋+ 1 is NP-Complete.

### On graphs double-critical with respect to the colouring number

The colouring number col($G$) of a graph $G$ is the smallest integer $k$ for which there is an ordering of the vertices of $G$ such that when removing the vertices of $G$ in the specified order no vertex of degree more than $k-1$ in the remaining graph is removed at any step. An edge $e$ of a graph $G$ is said to be double-col-critical if the colouring number of $G-V(e)$ is at most the colouring number of $G$ minus 2. A connected graph G is said to be double-col-critical if each edge of $G$ is double-col-critical. We characterise the double-col-critical graphs with colouring number at most 5. In addition, we prove that every 4-col-critical non-complete graph has at most half of its edges being double-col-critical, and that the extremal graphs are precisely the odd wheels on at least six vertices. We observe that for any integer $k$ greater than 4 and any positive number $ε$, there is a $k$-col-critical graph with the ratio of double-col-critical edges between $1- ε$ and 1.

### The game chromatic number of trees and forests

While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining in polynomial time the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4. This gives partial progress on the open question of the computational complexity of the game chromatic number of a forest.

### Snarks with total chromatic number 5

A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by χT(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with χT = 4 are said to be Type 1, and cubic graphs with χT = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n≥40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open.

### Maximum difference about the size of optimal identifying codes in graphs differing by one vertex

Let G=(V,E) be a simple undirected graph. We call any subset C⊆V an identifying code if the sets I(v)={c∈C | {v,c}∈E or v=c } are distinct and non-empty for all vertices v∈V. A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is denoted by γ(G). Let GS denote the induced subgraph of G where the vertex set S⊂V is deleted. We provide a tight upper bound for γ(GS)-γ(G) when both graphs are twin-free and |V| is large enough with respect to |S|. Moreover, we prove tight upper bound when G is a bipartite graph and |S|=1.

### Graphs with large disjunctive total domination number

Let G be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, γt(G). A set S of vertices in G is a disjunctive total dominating set of G if every vertex is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it. The disjunctive total domination number, γdt(G), is the minimum cardinality of such a set. We observe that γdt(G) ≤γt(G). Let G be a connected graph on n vertices with minimum degree δ. It is known [J. Graph Theory 35 (2000), 21 13;45] that if δ≥2 and n ≥11, then γt(G) ≤4n/7. Further [J. Graph Theory 46 (2004), 207 13;210] if δ≥3, then γt(G) ≤n/2. We prove that if δ≥2 and n ≥8, then γdt(G) ≤n/2 and we characterize the extremal graphs.

### Extending a perfect matching to a Hamiltonian cycle

Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.

### A conjecture on the number of Hamiltonian cycles on thin grid cylinder graphs

We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph $C_m \times P_{n+1}$. We distinguish two types of Hamiltonian cycles, and denote their numbers $h_m^A(n)$ and $h_m^B(n)$. For fixed $m$, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we derive their generating functions and other related results for $m\leq10$. The computational data we gathered suggests that $h^A_m(n)\sim h^B_m(n)$ when $m$ is even.

### On probe 2-clique graphs and probe diamond-free graphs

Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.

### p-box: a new graph model

In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

### Guarded subgraphs and the domination game

We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2-isometric subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are obtained, and then this concept is applied to the domination game. In this game two players, Dominator and Staller, alternate choosing vertices of a graph, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of Dominator is that the graph is dominated in as few steps as possible, while the aim of Staller is just the opposite. The game domination number is the number of vertices chosen when Dominator starts the game and both players play optimally. The main result of this paper is that the game domination number of a graph is not smaller than the game domination number of any guarded subgraph. Several applications of this result are presented.

### Edge stability in secure graph domination

A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

### Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes

If f is a binary word and d a positive integer, then the generalized Fibonacci cube Qd(f) is the graph obtained from the d-cube Qd by removing all the vertices that contain f as a factor, while the generalized Lucas cube Qd(lucas(f)) is the graph obtained from Qd by removing all the vertices that have a circulation containing f as a factor. The Fibonacci cube Γd and the Lucas cube Λd are the graphs Qd(11) and Qd(lucas(11)), respectively. It is proved that the connectivity and the edge-connectivity of Γd as well as of Λd are equal to ⌊ d+2 / 3⌋. Connected generalized Lucas cubes are characterized and generalized Fibonacci cubes are proved to be 2-connected. It is asked whether the connectivity equals minimum degree also for all generalized Fibonacci/Lucas cubes. It was checked by computer that the answer is positive for all f and all d≤9.

### Symmetric bipartite graphs and graphs with loops

We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges its two parts. To prove this, we study the relationship between symmetric bipartite graphs and graphs with loops.

### On the 1-2-3-conjecture

A k-edge-weighting of a graph G is a function w:E(G)→{1,…,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v∈V(G), c(v)=∑e∼vw(e). If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge-weighting (VC k-EW). Karoński et al. (J. Combin. Theory Ser. B, 91 (2004) 151 13;157) conjectured that every graph admits a VC 3-EW. This conjecture is known as the 1-2-3-conjecture. In this paper, first, we study the vertex-coloring edge-weighting of the Cartesian product of graphs. We prove that if the 1-2-3-conjecture holds for two graphs G and H, then it also holds for G□H. Also we prove that the Cartesian product of connected bipartite graphs admits a VC 2-EW. Moreover, we present several sufficient conditions for a graph to admit a VC 2-EW. Finally, we explore some bipartite graphs which do not admit a VC 2-EW.

### An approximability-related parameter on graphs―-properties and applications

We introduce a binary parameter on optimisation problems called separation. The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H-colourable subgraph (Max H-Col), which is a restriction of the general maximum constraint satisfaction problem (Max CSP) to a single, binary, and symmetric relation. Using known approximation ratios for Max k-cut, we obtain general asymptotic approximability results for Max H-Col for an arbitrary graph H. For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as Max H-Col.

### Ore-degree threshold for the square of a Hamiltonian cycle

A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n=2 contains a Hamiltonian cycle. In 1963, P´osa conjectured that every graph with minimum degree at least 2n=3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac’s theorem by proving that every graph with deg(u) + deg(v) ≥ n for every uv =2 E(G) contains a Hamiltonian cycle. Recently, Chˆau proved an Ore-type version of P´osa’s conjecture for graphs on n ≥ n0 vertices using the regularity–blow-up method; consequently the n0 is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller n0, we believe that our method of proof will be of independent interest.

### Generalized dynamic storage allocation

Dynamic Storage Allocation is a problem concerned with storing items that each have weight and time restrictions. Approximate algorithms have been constructed through online coloring of interval graphs. We present a generalization that uses online coloring of tolerance graphs. We utilize online-with-representation algorithms on tolerance graphs, which are online algorithms in which the corresponding tolerance representation of a vertex is also presented. We find linear bounds for the online-with-representation chromatic number of various classes of tolerance graphs and apply these results to a generalization of Dynamic Storage Allocation, giving us a polynomial time approximation algorithm with linear performance ratio.

### Toppling numbers of complete and random graphs

We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one chip to each neighbour; this may in turn cause other vertices to fire. The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the toppling number of a graph G, and is denoted by t(G). By considering strategies for both players and investigating the evolution of the game with differential equations, we provide asymptotic bounds on the toppling number of the complete graph. In particular, we prove that for sufficiently large n 0.596400 n2 < t(Kn) < 0.637152 n2. Using a fractional version of the game, we couple the toppling numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≥n² / √ log(n) asymptotically almost surely t(G(n,p))=(1+o(1)) p t(Kn).

### Partitioning the vertex set of a bipartite graph into complete bipartite subgraphs

Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove that the biclique vertex-partition problem is polynomially solvable for bipartite permutation graphs, bipartite distance-hereditary graphs and remains NP-complete for perfect elimination bipartite graphs and bipartite graphs containing no 4-cycles as induced subgraphs.

### Packing and covering the balanced complete bipartite multigraph with cycles and stars

Let Ck denote a cycle of length k and let Sk denote a star with k edges. For multigraphs F, G and H, an (F,G)-decomposition of H is an edge decomposition of H into copies of F and G using at least one of each. For L⊆H and R⊆rH, an (F,G)-packing (resp. (F,G)-covering) of H with leave L (resp. padding R) is an (F,G)-decomposition of H-E(L) (resp. H+E(R)). An (F,G)-packing (resp. (F,G)-covering) of H with the largest (resp. smallest) cardinality is a maximum (F,G)-packing (resp. minimum (F,G)-covering), and its cardinality is referred to as the (F,G)-packing number (resp. (F,G)-covering number) of H. In this paper, we determine the packing number and the covering number of λKn,n with Ck's and Sk's for any λ, n and k, and give the complete solution of the maximum packing and the minimum covering of λKn,n with 4-cycles and 4-stars for any λ and n with all possible leaves and paddings.

### Bounding the monomial index and (1,l)-weight choosability of a graph

Let G = (V,E) be a graph. For each e ∈E(G) and v ∈V(G), let Le and Lv, respectively, be a list of real numbers. Let w be a function on V(G) ∪E(G) such that w(e) ∈Le for each e ∈E(G) and w(v) ∈Lv for each v ∈V(G), and let cw be the vertex colouring obtained by cw(v) = w(v) + ∑ₑ ∋vw(e). A graph is (k,l)-weight choosable if there exists a weighting function w for which cw is proper whenever |Lv| ≥k and |Le| ≥l for every v ∈V(G) and e ∈E(G). A sufficient condition for a graph to be (1,l)-weight choosable was developed by Bartnicki, Grytczuk and Niwczyk (2009), based on the Combinatorial Nullstellensatz, a parameter which they call the monomial index of a graph, and matrix permanents. This paper extends their method to establish the first general upper bound on the monomial index of a graph, and thus to obtain an upper bound on l for which every admissible graph is (1,l)-weight choosable. Let ∂2(G) denote the smallest value s such that every induced subgraph of G has vertices at distance 2 whose degrees sum to at most s. We show that every admissible graph has monomial index at most ∂2(G) and hence that such graphs are (1, ∂2(G)+1)-weight choosable. While this does not improve the best known result on (1,l)-weight choosability, we show that the results can be extended to obtain improved bounds for some graph products; for instance, it is shown that G □ Kn is (1, nd+3)-weight choosable if G is d-degenerate.

### Genus distributions of cubic series-parallel graphs

We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are series-parallel graphs, this yields, by use of bar-amalgamation, a quadratic-time algorithm for every graph of treewidth at most 2 and maximum degree at most 3.

### Complexity of conditional colouring with given template

We study partitions of the vertex set of a given graph into cells that each induce a subgraph in a given family, and for which edges can have ends in different cells only when those cells correspond to adjacent vertices of a fixed template graph H. For triangle-free templates, a general collection of graph families for which the partitioning problem can be solved in polynomial time is described. For templates with a triangle, the problem is in some cases shown to be NP-complete.

### Oriented diameter and rainbow connection number of a graph

The oriented diameter of a bridgeless graph G is min diam(H) | H is a strang orientation of G. A path in an edge-colored graph G, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer number k for which there exists a k-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of rad(G) and η(G), where rad(G) is the radius of G and η(G) is the smallest integer number such that every edge of G is contained in a cycle of length at most η(G). We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph G in terms of the minimum degree of G.

### A four-sweep LBFS recognition algorithm for interval graphs

In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic Breadth-First Search (LBFS) and prove its correctness. They believe that their corresponding 5-sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4-sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.

### Balancedness of subclasses of circular-arc graphs

A graph is balanced if its clique-vertex incidence matrix contains no square submatrix of odd order with exactly two ones per row and per column. There is a characterization of balanced graphs by forbidden induced subgraphs, but no characterization by mininal forbidden induced subgraphs is known, not even for the case of circular-arc graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. In this work, we characterize when a given graph G is balanced in terms of minimal forbidden induced subgraphs, by restricting the analysis to the case where G belongs to certain classes of circular-arc graphs, including Helly circular-arc graphs, claw-free circular-arc graphs, and gem-free circular-arc graphs. In the case of gem-free circular-arc graphs, analogous characterizations are derived for two superclasses of balanced graphs: clique-perfect graphs and coordinated graphs.

### The generalized 3-connectivity of Lexicographic product graphs

The generalized k-connectivity κk(G) of a graph G, first introduced by Hager, is a natural generalization of the concept of (vertex-)connectivity. Denote by G^H and G&Box;H the lexicographic product and Cartesian product of two graphs G and H, respectively. In this paper, we prove that for any two connected graphs G and H, κ3(G^H)≥ κ3(G)|V(H)|. We also give upper bounds for κ3(G&Box; H) and κ3(G^H). Moreover, all the bounds are sharp.

### A variant of Niessen’s problem on degreesequences of graphs

Let (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) be two sequences of nonnegative integers satisfying the condition that b1>=b2>=...>=bn, ai<= bi for i=1,2,\textellipsis,n and ai+bi>=ai+1+bi+1 for i=1,2,\textellipsis, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) such that for every (c1,c2,\textellipsis,cn) with ai<=ci<=bi for i=1,2,\textellipsis,n and ∑&limits;i=1n ci=0 (mod 2), there exists a simple graph G with vertices v1,v2,\textellipsis,vn such that dG(vi)=ci for i=1,2,\textellipsis,n. This is a variant of Niessen\textquoterights problem on degree sequences of graphs (Discrete Math., 191 (1998), 247–253).

### On the Meyniel condition for hamiltonicity in bipartite digraphs

We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For a≥2, a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if d(u)+d(v)≥3a whenever uv∉A(D) and vu∉A(D). As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if δ(D)≥3a/2.

### On the Cartesian product of of an arbitrarily partitionable graph and a traceable graph

A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence τ=(n1,\textellipsis,nk) of positive integers that sum up to n, there exists a partition (V1,\textellipsis,Vk) of the vertex set V(G) such that each set Vi induces a connected subgraph of order ni. A graph G is called AP+1 if, given a vertex u∈V(G) and an index q∈ {1,\textellipsis,k}, such a partition exists with u∈Vq. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G□ H is AP+1. We also prove that G□H is AP, whenever G and H are AP and the order of one of them is not greater than four.

### The Price of Connectivity for Vertex Cover

The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the \em Price of Connectivity, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number t. We obtain forbidden induced subgraph characterizations for every real value t ≤q 3/2. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most t. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class […]

### The total irregularity of a graph

In this note a new measure of irregularity of a graph G is introduced. It is named the total irregularity of a graph and is defined as irr(t)(G) - 1/2 Sigma(u, v is an element of V(G)) vertical bar d(G)(u) - d(G)(v)vertical bar, where d(G)(u) denotes the degree of a vertex u is an element of V(G). All graphs with maximal total irregularity are determined. It is also shown that among all trees of the same order the star has the maximal total irregularity.

### Strong parity vertex coloring of plane graphs

A strong parity vertex coloring of a 2-connected plane graph is a coloring of the vertices such that every face is incident with zero or an odd number of vertices of each color. We prove that every 2-connected loopless plane graph has a strong parity vertex coloring with 97 colors. Moreover the coloring we construct is proper. This proves a conjecture of Czap and Jendrol' [Discuss. Math. Graph Theory 29 (2009), pp. 521-543.]. We also provide examples showing that eight colors may be necessary (ten when restricted to proper colorings).

### List circular backbone colouring

A natural generalization of graph colouring involves taking colours from a metric space and insisting that the endpoints of an edge receive colours separated by a minimum distance dictated by properties of the edge. In the q-backbone colouring problem, these minimum distances are either q or 1, depending on whether or not the edge is in the backbone. In this paper we consider the list version of this problem, with particular focus on colours in ℤp - this problem is closely related to the problem of circular choosability. We first prove that the list circular q-backbone chromatic number of a graph is bounded by a function of the list chromatic number. We then consider the more general problem in which each edge is assigned an individual distance between its endpoints, and provide bounds using the Combinatorial Nullstellensatz. Through this result and through structural approaches, we achieve good bounds when both the graph and the backbone belong to restricted families of graphs.

### Efficient open domination in graph products

A graph G is an efficient open domination graph if there exists a subset D of V(G) for which the open neighborhoods centered in vertices of D form a partition of V(G). We completely describe efficient open domination graphs among lexicographic, strong, and disjunctive products of graphs. For the Cartesian product we give a characterization when one factor is K2.

### On Hamiltonian Paths and Cycles in Sufficiently Large Distance Graphs

For a positive integer n∈ℕ and a set D⊆ ℕ, the distance graph GnD has vertex set { 0,1,\textellipsis,n-1} and two vertices i and j of GnD are adjacent exactly if |j-i|∈D. The condition gcd(D)=1 is necessary for a distance graph GnD being connected. Let D={d1,d2}⊆ℕ be such that d1>d2 and gcd(d1,d2)=1. We prove the following results. If n is sufficiently large in terms of D, then GnD has a Hamiltonian path with endvertices 0 and n-1. If d1d2 is odd, n is even and sufficiently large in terms of D, then GnD has a Hamiltonian cycle. If d1d2 is even and n is sufficiently large in terms of D, then GnD has a Hamiltonian cycle.

### On size, radius and minimum degree

Let G be a finite connected graph. We give an asymptotically tight upper bound on the size of G in terms of order, radius and minimum degree. Our result is a strengthening of an old classical theorem of Vizing (1967) if minimum degree is prescribed.

### The resolving number of a graph Delia

We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.

### 1-local 33/24-competitive Algorithm for Multicoloring Hexagonal Graphs

In the frequency allocation problem, we are given a cellular telephone network whose geographical coverage area is divided into cells, where phone calls are serviced by assigned frequencies, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e. minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph, where each vertex knows its position in the graph. We present a 1-local 33/24-competitive distributed algorithm for multicoloring a hexagonal graph, thereby improving the previous 1-local 7/5-competitive algorithm.

### A new characterization and a recognition algorithm of Lucas cubes

Fibonacci and Lucas cubes are induced subgraphs of hypercubes obtained by excluding certain binary strings from the vertex set. They appear as models for interconnection networks, as well as in chemistry. We derive a characterization of Lucas cubes that is based on a peripheral expansion of a unique convex subgraph of an appropriate Fibonacci cube. This serves as the foundation for a recognition algorithm of Lucas cubes that runs in linear time.

### Clique cycle transversals in graphs with few P₄'s

A graph is extended P4-laden if each of its induced subgraphs with at most six vertices that contains more than two induced P4's is 2K2,C4-free. A cycle transversal (or feedback vertex set) of a graph G is a subset T ⊆ V (G) such that T ∩ V (C) 6= ∅ for every cycle C of G; if, in addition, T is a clique, then T is a clique cycle transversal (cct). Finding a cct in a graph G is equivalent to partitioning V (G) into subsets C and F such that C induces a complete subgraph and F an acyclic subgraph. This work considers the problem of characterizing extended P4-laden graphs admitting a cct. We characterize such graphs by means of a finite family of forbidden induced subgraphs, and present a linear-time algorithm to recognize them.

### Maximal independent sets in bipartite graphs with at least one cycle

A maximal independent set is an independent set that is not a proper subset of any other independent set. Liu [J.Q. Liu, Maximal independent sets of bipartite graphs, J. Graph Theory, 17 (4) (1993) 495-507] determined the largest number of maximal independent sets among all n-vertex bipartite graphs. The corresponding extremal graphs are forests. It is natural and interesting for us to consider this problem on bipartite graphs with cycles. Let \mathscrBₙ (resp. \mathscrBₙ') be the set of all n-vertex bipartite graphs with at least one cycle for even (resp. odd) n. In this paper, the largest number of maximal independent sets of graphs in \mathscrBₙ (resp. \mathscrBₙ') is considered. Among \mathscrBₙ the disconnected graphs with the first-, second-, \ldots, \fracn-22-th largest number of maximal independent sets are characterized, while the connected graphs in \mathscrBₙ having the largest, the second largest number of maximal independent sets are determined. Among \mathscrBₙ' graphs have the largest number of maximal independent sets are identified.

### A note on the NP-hardness of two matching problems in induced subgrids

Given a graph, finding the maximal matching of minimum size (MMM) and the induced matching of maximum size (MIM) have been very popular research topics during the last decades. In this paper, we give new complexity results, namely the NP-hardness of MMM and MIM in induced subgrids and we point out some promising research directions. We also sketch the general framework of a unified approach to show the NP-hardness of some problems in subgrids.

### Probe interval graphs and probe unit interval graphs on superclasses of cographs

A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe (unit) interval graphs form a superclass of (unit) interval graphs. Probe interval graphs were introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. The main results of this article are minimal forbidden induced subgraphs characterizations of probe interval and probe unit interval graphs within two superclasses of cographs: P4-tidy graphs and tree-cographs. Furthermore, we introduce the concept of graphs class with a companion which allows to describe all the minimally non-(probe G) graphs with disconnected complement for every graph class G with a companion.

### Removable edges in near-bricks

For a brick apart from a few small graphs, Lovász (1987) proposed a conjecture on the existence of an edge whose deletion results in a graph with only one brick in its tight cut decomposition. Carvalho, Lucchesi, and Murty (2002) confirmed this conjecture by showing the existence of such two edges. This paper generalizes the result obtained by Carvalho et al. to the case of irreducible near-brick, where a graph is irreducible if it contains no induced odd path of length 3 or more. Meanwhile, a lower bound on the number of removable edges of matching-covered bipartite graphs is presented.

### Improved bounds on the crossing number of butterfly network

We draw the r-dimensional butterfly network with 1 / 44r+O(r2r) crossings which improves the previous estimate given by Cimikowski (1996). We also give a lower bound which matches the upper bound obtained in this paper.

### Bipartite powers of k-chordal graphs

Let k be an integer and k ≥3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if Gm is chordal then so is Gm+2. Brandstädt et al. in [Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if Gm is k-chordal, then so is Gm+2. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. The m-th bipartite power G[m] of a bipartite graph G is the bipartite graph obtained from G by adding edges (u,v) where dG(u,v) is odd and less than or equal to m. Note that G[m] = G[m+1] for each odd m. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G[m], where k, m are positive integers with k≥4.

### The b-chromatic number of powers of cycles

A b-coloring of a graph G by k colors is a proper vertex coloring such that each color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number χb(G) is the maximum integer k for which G has a b-coloring by k colors. Let Cnr be the rth power of a cycle of order n. In 2003, Effantin and Kheddouci established the b-chromatic number χb(Cnr) for all values of n and r, except for 2r+3≤n≤3r. For the missing cases they presented the lower bound L:= min n-r-1,r+1+⌊ n-r-1 / 3⌋ and conjectured that χb(Cnr)=L. In this paper, we determine the exact value on χb(Cnr) for the missing cases. It turns out that χb(Cnr)>L for 2r+3≤n≤2r+3+r-6 / 4.

### All totally symmetric colored graphs

In this paper we describe all edge-colored graphs that are fully symmetric with respect to colors and transitive on every set of edges of the same color. They correspond to fully symmetric homogeneous factorizations of complete graphs. Our description completes the work done in our previous paper, where we have shown, in particular, that there are no such graphs with more than 5 colors. Using some recent results, with a help of computer, we settle all the cases that was left open in the previous paper.

### A chip-firing variation and a new proof of Cayley's formula

We introduce a variation of chip-firing games on connected graphs. These 'burn-off' games incorporate the loss of energy that may occur in the physical processes that classical chip-firing games have been used to model. For a graph G=(V,E), a configuration of 'chips' on its nodes is a mapping C:V→ℕ. We study the configurations that can arise in the course of iterating a burn-off game. After characterizing the 'relaxed legal' configurations for general graphs, we enumerate the 'legal' ones for complete graphs Kn. The number of relaxed legal configurations on Kn coincides with the number tn+1 of spanning trees of Kn+1. Since our algorithmic, bijective proof of this fact does not invoke Cayley's Formula for tn, our main results yield secondarily a new proof of this formula.

### Krausz dimension and its generalizations in special graph classes

A Krausz (k,m)-partition of a graph G is a decomposition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-Krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a Krausz (k,m)-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤3" is polynomially solvable for chordal graphs, thus partially solving the open problem of P. Hlineny and J. Kratochvil. We solve another open problem of P. Hlineny and J. Kratochvil by proving that the problem of finding Krausz dimension is NP-hard for split graphs and complements of bipartite graphs. We show that the problem of finding m-Krausz dimension is NP-hard for every m≥1, but the problem "kdimm(G)≤k" is is fixed-parameter tractable when parameterized by k and m for (∞,1)-polar graphs. Moreover, the class of (∞,1)-polar graphs with kdimm(G)≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.

### Further results on maximal nontraceable graphs of smallest size

Let g(n) denote the minimum number of edges of a maximal nontraceable (MNT) graph of order n. In 2005 Frick and Singleton (Lower bound for the size of maximal nontraceable graphs, Electronic Journal of Combinatorics, 12(1) R32, 2005) proved that g(n) = ⌈3n-22 ⌉ for n ≥54 as well as for n ∈I, where I= 12,13,22,23,30,31,38,39, 40,41,42,43,46,47,48,49,50,51 and they determined g(n) for n ≤9. We determine g(n) for 18 of the remaining 26 values of n, showing that g(n) = ⌈ 3n-22 ⌉ for n ≥54 as well as for n ∈I ∪18,19,20,21,24,25,26,27,28, 29,32,33 and g(n) = ⌈ 3n2 ⌉ for n ∈ 10, 11, 14, 15, 16, 17. We give results based on ''analytic'' proofs as well as computer searches.

### The determining number of Kneser graphs

A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G is the minimum cardinality of a determining set of G. This paper studies the determining number of Kneser graphs. First, we compute the determining number of a wide range of Kneser graphs, concretely Kn:k with n≥k(k+1) / 2+1. In the language of group theory, these computations provide exact values for the base size of the symmetric group Sn acting on the k-subsets of 1,..., n. Then, we establish for which Kneser graphs Kn:k the determining number is equal to n-k, answering a question posed by Boutin. Finally, we find all Kneser graphs with fixed determining number 5, extending the study developed by Boutin for determining number 2, 3 or 4.

### Sequence variations of the 1-2-3 conjecture and irregularity strength

Karonski, Luczak, and Thomason (2004) conjecture that, for any connected graph G on at least three vertices, there exists an edge weighting from 1, 2, 3 such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) make a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than 1, 2, 3. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. Such a colouring relies on an ordering of E(G), and so two variations arise - one where we may choose any ordering of E(G) and one where the ordering is fixed. In the former case, we bound the list size required for any graph. In the latter, we obtain a bound on list sizes for graphs with sufficiently large minimum degree. We also extend our methods to a list variation of irregularity strength, where each vertex receives a distinct sequence of edge weights.

### Upper k-tuple domination in graphs

For a positive integer k, a k-tuple dominating set of a graph G is a subset S of V (G) such that |N [v] ∩ S| ≥ k for every vertex v, where N [v] = {v} ∪ {u ∈ V (G) : uv ∈ E(G)}. The upper k-tuple domination number of G, denoted by Γ×k (G), is the maximum cardinality of a minimal k-tuple dominating set of G. In this paper we present an upper bound on Γ×k (G) for r-regular graphs G with r ≥ k, and characterize extremal graphs achieving the upper bound. We also establish an upper bound on Γ×2 (G) for claw-free r-regular graphs. For the algorithmic aspect, we show that the upper k-tuple domination problem is NP-complete for bipartite graphs and for chordal graphs.

### Secure frameproof codes through biclique covers

For a binary code Γ of length v, a v-word w produces by a set of codewords {w1,...,wr}⊆Γ if for all i=1,...,v, we have wi∈{w1i,...,wri} . We call a code r-secure frameproof of size t if |Γ|=t and for any v-word that is produced by two sets C1 and C2 of size at most r then the intersection of these sets is nonempty. A d-biclique cover of size v of a graph G is a collection of v-complete bipartite subgraphs of G such that each edge of G belongs to at least d of these complete bipartite subgraphs. In this paper, we show that for t≥2r, an r-secure frameproof code of size t and length v exists if and only if there exists a 1-biclique cover of size v for the Kneser graph KG(t,r) whose vertices are all r-subsets of a t-element set and two r-subsets are adjacent if their intersection is empty. Then we investigate some connection between the minimum size of d-biclique covers of Kneser graphs and cover-free families, where an (r,w;d) cover-free family is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. Also, we present an upper bound for 1-biclique covering number of Kneser graphs.

### A note on planar Ramsey numbers for a triangle versus wheels

For two given graphs G and H , the planar Ramsey number P R ( G; H ) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G , or its complement contains a copy of H . In this paper, we determine all planar Ramsey numbers for a triangle versus wheels.

### On 4-valent Frobenius circulant graphs

A 4-valent first-kind Frobenius circulant graph is a connected Cayley graph DLn(1, h) = Cay(Zn, H) on the additive group of integers modulo n, where each prime factor of n is congruent to 1 modulo 4 and H = {[1], [h], −[1], −[h]} with h a solution to the congruence equation x 2 + 1 ≡ 0 (mod n). In [A. Thomson and S. Zhou, Frobenius circulant graphs of valency four, J. Austral. Math. Soc. 85 (2008), 269-282] it was proved that such graphs admit 'perfect ' routing and gossiping schemes in some sense, making them attractive candidates for modelling interconnection networks. In the present paper we prove that DLn(1, h) has the smallest possible broadcasting time, namely its diameter plus two, and we explicitly give an optimal broadcasting in DLn(1, h). Using number theory we prove that it is possible to recursively construct larger 4-valent first-kind Frobenius circulants from smaller ones, and we give a methodology for such a construction. These and existing results suggest that, among all 4-valent circulant graphs, 4-valent first-kind Frobenius circulants are extremely efficient in terms of routing, gossiping, broadcasting and recursive construction.

### Immersion containment and connectivity in color-critical graphs

The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. It is shown that a t-chromatic graph G contains either an immersed Kt or an immersed t-chromatic subgraph that is both 4-vertex-connected and t-edge-connected. This gives supporting evidence of our conjecture that if G requires at least t colors, then Kt is immersed in G.

### Acyclic chromatic index of fully subdivided graphs and Halin graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). A graph G is called fully subdivided if it is obtained from another graph H by replacing every edge by a path of length at least two. Fully subdivided graphs are known to be acyclically edge colorable using Δ+1 colors since they are properly contained in 2-degenerate graphs which are acyclically edge colorable using Δ+1 colors. Muthu, Narayanan and Subramanian gave a simple direct proof of this fact for the fully subdivided graphs. Fiamcik has shown that if we subdivide every edge in a cubic graph with at most two exceptions to get a graph G, then a'(G)=3. In this paper we generalise the bound to Δ for all fully subdivided graphs improving the result of Muthu et al. In particular, we prove that if G is a fully subdivided graph and Δ(G) ≥3, then a'(G)=Δ(G). Consider a graph G=(V,E), with E=E(T) ∪E(C) where T is a rooted tree on the vertex set V and C is a simple cycle on the leaves of T. Such a graph G is called a Halin graph if G has a planar embedding and T has no vertices of degree 2. Let Kn denote a complete graph on n vertices. Let G be a Halin graph with maximum degree Δ. We prove that, a'(G) = 5 if G is K4, 4 if Δ = 3 and G is not K4, and Δ otherwise.

### Random Cayley digraphs of diameter 2 and given degree

We consider random Cayley digraphs of order n with uniformly distributed generating sets of size k. Specifically, we are interested in the asymptotics of the probability that such a Cayley digraph has diameter two as n -> infinity and k = f(n), focusing on the functions f(n) = left perpendicularn(delta)right perpendicular and f(n) = left perpendicularcnright perpendicular. In both instances we show that this probability converges to 1 as n -> infinity for arbitrary fixed delta is an element of (1/2, 1) and c is an element of (0, 1/2), respectively, with a much larger convergence rate in the second case and with sharper results for Abelian groups.

### Graphs with many vertex-disjoint cycles

We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

### On paths, trails and closed trails in edge-colored graphs

In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since […]

### The competition number of a generalized line graph is at most two

In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.

### On bipartite powers of bigraphs

The notion of graph powers is a well-studied topic in graph theory and its applications. In this paper, we investigate a bipartite analogue of graph powers, which we call bipartite powers of bigraphs. We show that the classes of bipartite permutation graphs and interval bigraphs are closed under taking bipartite power. We also show that the problem of recognizing bipartite powers is NP-complete in general.

### alpha-Labelings and the Structure of Trees with Nonzero alpha-Deficit

We present theoretical and computational results on alpha-labelings of trees. The theorems proved in this paper were inspired by the results of a computer investigation of alpha-labelings of all trees with up to 26 vertices, all trees with maximum degree 3 and up to 36 vertices, all trees with maximum degree 4 and up to 32 vertices and all trees with maximum degree 5 and up to 31 vertices. We generalise a criterion for trees to have nonzero alpha-deficit, and prove an unexpected result on the alpha-deficit of trees with a vertex of large degree compared to the order of the tree.

### The generalized 3-connectivity of Cartesian product graphs

The generalized connectivity of a graph, which was introduced by Chartrand et al. in 1984, is a generalization of the concept of vertex connectivity. Let S be a nonempty set of vertices of G, a collection \T-1, T (2), ... , T-r\ of trees in G is said to be internally disjoint trees connecting S if E(T-i) boolean AND E(T-j) - empty set and V (T-i) boolean AND V(T-j) = S for any pair of distinct integers i, j, where 1 <= i, j <= r. For an integer k with 2 <= k <= n, the k-connectivity kappa(k) (G) of G is the greatest positive integer r for which G contains at least r internally disjoint trees connecting S for any set S of k vertices of G. Obviously, kappa(2)(G) = kappa(G) is the connectivity of G. Sabidussi's Theorem showed that kappa(G square H) >= kappa(G) + kappa(H) for any two connected graphs G and H. In this paper, we prove that for any two connected graphs G and H with kappa(3) (G) >= kappa(3) (H), if kappa(G) > kappa(3) (G), then kappa(3) (G square H) >= kappa(3) (G) + kappa(3) (H); if kappa(G) = kappa(3)(G), then kappa(3)(G square H) >= kappa(3)(G) + kappa(3) (H) - 1. Our result could be seen as an extension of Sabidussi's Theorem. Moreover, all the bounds are sharp.

### Nordhaus-Gaddum Type Results for Total Domination

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we study Nordhaus-Gaddum-type results for total domination. We examine the sum and product of γt(G1) and γt(G2) where G1 ⊕G2 = K(s,s), and γt is the total domination number. We show that the maximum value of the sum of the total domination numbers of G1 and G2 is 2s+4, with equality if and only if G1 = sK2 or G2 = sK2, while the maximum value of the product of the total domination numbers of G1 and G2 is max{8s,⌊(s+6)2/4 ⌋}.

### On the number of factors in codings of three interval exchange

We consider exchange of three intervals with permutation (3, 2, 1). The aim of this paper is to count the cardinality of the set 3iet (N) of all words of length N which appear as factors in infinite words coding such transformations. We use the strong relation of 3iet words and words coding exchange of two intervals, i.e., Sturmian words. The known asymptotic formula #2iet(N)/N-3 similar to 1/pi(2) for the number of Sturmian factors allows us to find bounds 1/3 pi(2) +o(1) \textless= #3iet(N)N-4 \textless= 2 pi(2) + o(1)