# Vol. 16 no. 1

### 1. 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.

### 2. 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).

### 3. 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.
Section: Graph Theory

### 4. 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 &#x007b; 0,1,\textellipsis,n-1&#x007d; 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=&#x007b;d1,d2&#x007d;⊆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.
Section: Graph Theory

### 5. The Price of Mediation

We study the relationship between correlated equilibria and Nash equilibria. In contrast to previous work focusing on the possible benefits of a benevolent mediator, we define and bound the Price of Mediation (PoM): the ratio of the social cost (or utility) of the worst correlated equilibrium to the social cost (or utility) of the worst Nash. We observe that in practice, the heuristics used for mediation are frequently non-optimal, and from an economic perspective mediators may be inept or self-interested. Recent results on computation of equilibria also motivate our work. We consider the Price of Mediation for general games with small numbers of players and pure strategies. For two player, two strategy games we give tight bounds in the non-negative cost model and the non-negative utility model. For larger games (either more players, or more pure strategies per player, or both) we show that the PoM can be arbitrary. We also have many results on symmetric congestion games (also known as […]
Section: Discrete Algorithms

### 6. Descents after maxima in compositions

We consider compositions of n, i.e., sequences of positive integers (or parts) (σi)i=1k where σ1+σ2+...+σk=n. We define a maximum to be any part which is not less than any other part. The variable of interest is the size of the descent immediately following the first and the last maximum. Using generating functions and Mellin transforms, we obtain asymptotic expressions for the average size of these descents. Finally, we show with the use of a simple bijection between the compositions of n for n>1, that on average the descent after the last maximum is greater than the descent after the first.
Section: Combinatorics

### 7. Graphs where every k-subset of vertices is an identifying set

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these intersections are different for different vertices $x$. Let $k$ be a positive integer. We will consider graphs where \emph{every} $k$-subset is identifying. We prove that for every $k>1$ the maximal order of such a graph is at most $2k-2.$ Constructions attaining the maximal order are given for infinitely many values of $k.$ The corresponding problem of $k$-subsets identifying any at most $\ell$ vertices is considered as well.
Section: Combinatorics

### 8. 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.
Section: Graph Theory

### 9. 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.
Section: Graph Theory

### 10. Computation with No Memory, and Rearrangeable Multicast Networks

We investigate the computation of mappings from a set S^n to itself with "in situ programs", that is using no extra variables than the input, and performing modifications of one component at a time, hence using no extra memory. In this paper, we survey this problem introduced in previous papers by the authors, we detail its close relation with rearrangeable multicast networks, and we provide new results for both viewpoints. A bijective mapping can be computed by 2n-1 component modifications, that is by a program of length 2n-1, a result equivalent to the rearrangeability of the concatenation of two reversed butterfly networks. For a general arbitrary mapping, we give two methods to build a program with maximal length 4n-3. Equivalently, this yields rearrangeable multicast routing methods for the network formed by four successive butterflies with alternating reversions. The first method is available for any set S and practically equivalent to a known method in network theory. […]

The implicit signature κ consists of the multiplication and the (ω-1)-power. We describe a procedure to transform each κ-term over a finite alphabet A into a certain canonical form and show that different canonical forms have different interpretations over some finite semigroup. The procedure of construction of the canonical forms, which is inspired in McCammond\textquoterights normal form algorithm for ω-terms interpreted over the pseudovariety A of all finite aperiodic semigroups, consists in applying elementary changes determined by an elementary set Σ of pseudoidentities. As an application, we deduce that the variety of κ-semigroups generated by the pseudovariety S of all finite semigroups is defined by the set Σ and that the free κ-semigroup generated by the alphabet A in that variety has decidable word problem. Furthermore, we show that each ω-term has a unique ω-term in canonical form with the same value over A. In particular, the canonical forms provide new, simpler, […]
Section: Automata, Logic and Semantics

### 12. 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).
Section: Graph Theory

### 13. A Parameterized Measure-and-ConquerAnalysis for Finding a k-Leaf Spanning Treein an Undirected Graph

The problem of finding a spanning tree in an undirected graph with a maximum number of leaves is known to be NP-hard. We present an algorithm which finds a spanning tree with at least k leaves in time O*(3.4575k) which improves the currently best algorithm. The estimation of the running time is done by using a non-standard measure. The present paper is one of the still few examples that employ the Measure & Conquer paradigm of algorithm analysis in the area of Parameterized Algorithmics.
Section: Discrete Algorithms

### 14. 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.
Section: Graph Theory

### 15. 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∈ &#x007b;1,\textellipsis,k&#x007d;, 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.
Section: Graph Theory

### 16. 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 […]
Section: Graph Theory

### 17. An S-adic characterization of minimal subshifts with first difference of complexity 1 ≤ p(n+1) - p(n) ≤ 2

An S-adic characterization of minimal subshifts with first difference of complexity 1 ≤ p(n + 1) − p(n) ≤ 2 S. Ferenczi proved that any minimal subshift with first difference of complexity bounded by 2 is S-adic with Card(S) ≤ 3 27. In this paper, we improve this result by giving an S-adic characterization of these subshifts with a set S of 5 morphisms, solving by this way the S-adic conjecture for this particular case.
Section: Automata, Logic and Semantics

### 18. 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 &#x2211;&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&#x2013;253).
Section: Graph Theory

### 19. 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&#x2265;2, a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if d(u)+d(v)&#x2265;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)&#x2265;3a/2.
Section: Graph Theory

### 20. Computing the number of h-edge spanning forests in complete bipartite graphs

Let fm,n,h be the number of spanning forests with h edges in the complete bipartite graph Km,n. Kirchhoff\textquoterights Matrix Tree Theorem implies fm,n,m+n-1=mn-1 nm-1 when m &#x2265;1 and n &#x2265;1, since fm,n,m+n-1 is the number of spanning trees in Km,n. In this paper, we give an algorithm for computing fm,n,h for general m,n,h. We implement this algorithm and use it to compute all non-zero fm,n,h when m &#x2264;50 and n &#x2264;50 in under 2 days.
Section: Analysis of Algorithms

### 21. Elements in finite classical groups whose powers have large 1-Eigenspaces

We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.

### 22. Congruence successions in compositions

A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.
Section: Combinatorics

### 23. A combinatorial non-commutative Hopf algebra of graphs

A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.
Section: Combinatorics

### 24. 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)&#x2265; κ3(G)|V(H)|. We also give upper bounds for κ3(G&Box; H) and κ3(G^H). Moreover, all the bounds are sharp.
Section: Graph Theory

### 25. Robot's hand and expansions in non-integer bases

We study a robot hand model in the framework of the theory of expansions in non-integer bases. We investigate the reachable workspace and we study some configurations enjoying form closure properties. Keywords: Robot hand, discrete control, expansions in non-integer bases, expansions in complex bases