Related papers: Local girth choosability of planar graphs

The local metric dimension ${\rm dim}_l$ in relation to the clique number $\omega$ is investigated. It is proved that if $\omega(G)\leq n(G)-3$, then ${\rm dim}_l(G) \leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the…

This note proves that every graph of Euler genus $\mu$ is $\lceil 2 + \sqrt{3\mu + 3}\,\rceil$--choosable with defect 1 (that is, clustering 2). Thus, allowing defect as small as 1 reduces the choice number of surface embeddable graphs…

An $(L,d)^*$-coloring is a mapping $\phi$ that assigns a color $\phi(v)\in L(v)$ to each vertex $v\in V(G)$ such that at most $d$ neighbors of $v$ receive colore $\phi(v)$. A graph is called $(m,d)^*$-choosable, if $G$ admits an…

We prove the following local strengthening of Shearer's classic bound on the independence number of triangle-free graphs: For every triangle-free graph $G$ there exists a probability distribution on its independent sets such that every…

For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total…

For a graph $G$ with a list assignment $L$ and two $L$-colorings $\alpha$ and $\beta$, an $L$-recoloring sequence from $\alpha$ to $\beta$ is a sequence of proper $L$-colorings where consecutive colorings differ at exactly one vertex. We…

We study the linear list chromatic number, denoted $\lcl(G)$, of sparse graphs. The maximum average degree of a graph $G$, denoted $\mad(G)$, is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with…

The choosability $\chi_\ell(G)$ of a graph $G$ is the minimum $k$ such that having $k$ colors available at each vertex guarantees a proper coloring. Given a toroidal graph $G$, it is known that $\chi_\ell(G)\leq 7$, and $\chi_\ell(G)=7$ if…

Xu and Wu proved that if every $5$-cycle of a planar graph $G$ is not simultaneously adjacent to $3$-cycles and $4$-cycles, then $G$ is $4$-choosable. In this paper, we improve this result as follows. Let $\{i, j, k, l\} = \{3,4,5,6\}.$ For…

A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A characterization of $3$-choice critical graphs was given by Voigt in [On list Colourings and Choosability of Graphs,…

Suppose $G$ is a graph and $L$ is a list assignment for $G$. A request of $L$ is a function $r$ with nonempty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$. The triple $(G,L,r)$ is $\epsilon$-satisfiable if there…

We show that every graph with two crossings is 5-choosable. We also prove that every graph which can be made planar by removing one edge is 5-choosable.

We prove that every triangle-free $4$-critical graph $G$ satisfies $e(G) \geq \frac{5v(G)+2}{3}$. This result gives a unified proof that triangle-free planar graphs are $3$-colourable, and that graphs of girth at least five which embed in…

We consider the flexible list coloring problem, in which we have a graph $G$, a color list assignment $L:V(G) \rightarrow 2^{\mathbb N}$, and a set $U \subseteq V(G)$ of vertices such that each $u \in U$ has a preferred color $p(u) \in…

A proper conflict-free coloring of a graph is a proper vertex coloring wherein each non-isolated vertex's open neighborhood contains at least one color appearing exactly once. For a non-negative integer $k$, a graph $G$ is said to be proper…

The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds…

This paper proves that every planar graph $G$ contains a matching $M$ such that the Alon-Tarsi number of $G-M$ is at most $4$. As a consequence, $G-M$ is $4$-paintable, and hence $G$ itself is $1$-defective $4$-paintable. This improves a…

Thomassen famously proved that every planar graph is 5-choosable. We explore variants of this result, focusing on finding disjoint correspondence colorings, in the more general class of $K_5$-minor-free graphs. Correspondence colorings…

The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph $G$ has as vertices all possible $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex. We use a list coloring technique…

Let $a,b$ be positive integers with $a\ge b$. A graph $G$ is $(a,b)$-choosable if, for every assignment of lists $L(v)$ of size $a$ to the vertices of $G$, there exists a choice of subsets $C(v)\subseteq L(v)$ with $|C(v)|=b$ for each $v$…