# 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.

### 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 […]

### 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 ≥ […]

### On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance

Assume that $n, \delta ,k$ are integers with $0 \leq k < \delta < n$. Given a graph $G=(V,E)$ with $|V|=n$. The symbol $G-F, F \subseteq V$, denotes the graph with $V(G-F)=V-F$, and $E(G-F)$ obtained by $E$ after deleting the edges with at least one endvertex in $F$. $G$ is called $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, or $k$-vertex fault Hamiltonian-connected if $G-F$ remains traceable, Hamiltonian, and Hamiltonian-connected for all $F$ with $0 \leq |F| \leq k$, respectively. The notations $h_1(n, \delta ,k)$, $h_2(n, \delta ,k)$, and $h_3(n, \delta ,k)$ denote the minimum number of edges required to guarantee an $n$-vertex graph with minimum degree $\delta (G) \geq \delta$ to be $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, and $k$-vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the $k$-vertex fault […]

### Edge Disjoint Hamilton Cycles in Knödel Graphs

The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $\lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.

### Traceability of locally hamiltonian and locally traceable graphs

If $\mathcal{P}$ is a given graph property, we say that a graph $G$ is locally $\mathcal{P}$ if $\langle N(v) \rangle$ has property $\mathcal{P}$ for every $v \in V(G)$ where $\langle N(v) \rangle$ is the induced graph on the open neighbourhood of the vertex $v$. Pareek and Skupien (C. M. Pareek and Z. Skupien , On the smallest non-Hamiltonian locally Hamiltonian graph, J. Univ. Kuwait (Sci.), 10:9 - 17, 1983) posed the following two questions. Question 1 Is 9 the smallest order of a connected nontraceable locally traceable graph? Question 2 Is 14 the smallest order of a connected nontraceable locally hamiltonian graph? We answer the second question in the affirmative, but show that the correct number for the first question is 10. We develop a technique to construct connected locally hamiltonian and locally traceable graphs that are not traceable. We use this technique to construct such graphs with various prescribed properties.