Related papers: Variable degeneracy of planar graphs without chord…

We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if $H$ is a fixed planar graph that has a planar embedding with all the vertices with degree at least 4 on a single…

For any positive integer $k$, the reconfiguration graph for all $k$-colorings of a graph $G$, denoted by $\mathcal{R}_k(G)$, is the graph where vertices represent the $k$-colorings of $G$, and two $k$-colorings are joined by an edge if they…

Given an infinite family ${\mathcal G}$ of graphs and a monotone property ${\mathcal P}$, an (upper) threshold for ${\mathcal G}$ and ${\mathcal P}$ is a "fastest growing" function $p: \mathbb{N} \to [0,1]$ such that $\lim_{n \to \infty}…

Let $G$ be a graph. We say that $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. We use $P_t$ and $C_t$ to denote a path…

A sequence $s_1,s_2,...,s_k,s_1,s_2,...,s_k$ is a repetition. A sequence $S$ is nonrepetitive, if no subsequence of consecutive terms of $S$ form a repetition. Let $G$ be a vertex colored graph. A path of $G$ is nonrepetitive, if the…

Odd coloring is a variant of proper coloring and has received wide attention. We study the list-coloring version of this notion in this paper. We prove that if $G$ is a graph embeddable in the torus or the Klein bottle with no cycle of…

It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs,…

In the flexible list coloring problem, we consider a graph $G$ and a color list assignment $L$ on $G$, as well as a subset $U \subseteq V(G)$ for which each $u \in U$ has a preferred color $p(u) \in L(u)$. Our goal is to find a proper…

In 2012, L\'ev\^eque, Maffray, and Trotignon conjectured that each graph $G$ that contains no induced subdivision of $K_4$ is $4$-colorable. In this paper, we prove that this conjecture holds when $G$ contains a $K_{1,2,3}$.

A graph is $(d_1, ..., d_r)$-colorable if its vertex set can be partitioned into $r$ sets $V_1, ..., V_r$ so that the maximum degree of the graph induced by $V_i$ is at most $d_i$ for each $i\in \{1, ..., r\}$. For a given pair $(g, d_1)$,…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$…

A graph $H$ is an immersion of a graph $G$ if $H$ can be obtained by some sugraph $G$ after lifting incident edges. We prove that there is a polynomial function $f:\Bbb{N}\times\Bbb{N}\rightarrow\Bbb{N}$, such that if $H$ is a connected…

Eigenvectors of the Laplacian of a cycle graph exhibit the sinusoidal characteristics of the standard DFT basis, and signals defined on such graphs are amenable to linear shift invariant (LSI) operations. In this paper we propose to reduce…

If $f : \tilde{\Gamma} \rightarrow \Gamma$ is a covering map between connected graphs, and $H$ is the subgroup of $\pi_1(\Gamma,v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is…

A coloring of a graph $G=(V,E)$ is a partition $\{V_1, V_2, \ldots, V_k\}$ of $V$ into independent sets or color classes. A vertex $v\in V_i$ is a Grundy vertex if it is adjacent to at least one vertex in each color class $V_j$ for every…

We obtain some $d\ge2$ such that every graph $G$ with no induced copy of the five-vertex path $P_5$ has at most $\alpha(G)\omega(G)^d$ vertices. This ``off-diagonal Ramsey'' statement implies that every such graph $G$ has fractional…

A {\em hole} is an induced cycle of length at least 4, a $k$-hole is a hole of length $k$, and an {\em odd hole} is a hole of odd length. Let $\ell\ge 2$ be an integer. Let ${\cal A}_{\ell}$ be the family of graphs of girth at least $2\ell$…

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$,…

Let $G$ and $H$ be two vertex disjoint graphs. The {\em union} $G\cup H$ is the graph with $V(G\cup H)=V(G)\cup (H)$ and $E(G\cup H)=E(G)\cup E(H)$. The {\em join} $G+H$ is the graph with $V(G+H)=V(G)+V(H)$ and $E(G+H)=E(G)\cup…

Given a family $\mathcal{H}$ of graphs, a graph $G$ is called $\mathcal{H}$-universal if $G$ contains every graph of $\mathcal{H}$ as a subgraph. Following the extensive research on universal graphs of small size for bounded-degree graphs,…