Related papers: List $4$-colouring of planar graphs

This paper proves the following result: Assume $G$ is a triangle free planar graph, $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $\mid L(v)\mid = 4$ for each vertex $v \in V(G)-X$ and $\mid L(v)\mid = 3$…

A list assignment $L$ for a graph $G$ is an $(\ell,k)$-list assignment if $|L(v)|\geq \ell$ for each $v \in V(G)$ and $|L(u) \cap L(v)| \leq k$ for each $uv \in E(G)$. We say $G$ is $(\ell,k)$-choosable if it admits an $L$-colouring for…

It is known that every loopless cubic graph is 4-edge choosable. We prove the following strengthened result. Let G be a planar cubic graph having b cut-edges. There exists a set F of at most 5b/2 edges of G with the following property. For…

Let $G$ be a $\{C_4, C_5\}$-free planar graph with a list assignment $L$. Suppose a preferred color is given for some of the vertices. We prove that if all lists have size at least four, then there exists an $L$-coloring respecting at least…

A graph where each vertex $v$ has a list $L(v)$ of available colors is $L$-colorable if there is a proper coloring such that the color of $v$ is in $L(v)$ for each $v$. A graph is $k$-choosable if every assignment $L$ of at least $k$ colors…

Let $F$ be a (possibly improper) edge-coloring of a graph $G$; a vertex coloring of $G$ is \emph{adapted to} $F$ if no color appears at the same time on an edge and on its two endpoints. If for some integer $k$, a graph $G$ is such that…

A (k,d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k,d)-choosable if there exists an…

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms…

Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G is triangle-free and all lists have size at least four, then there exists an L-coloring respecting at least a…

Thomassen showed that planar graphs are 5-list-colourable, and that planar graphs of girth at least five are 3-list-colourable. An easy degeneracy argument shows that planar graphs of girth at least four are 4-list-colourable. In 2022,…

A list assignment of a graph $G$ is a function $L$ that assigns a list $L(v)$ of colors to each vertex $v\in V(G)$. An $(L,d)^*$-coloring is a mapping $\pi$ that assigns a color $\pi(v)\in L(v)$ to each vertex $v\in V(G)$ so that at most…

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvo\v{r}\'ak and Postle (2017). In this paper, we prove that every planar graph $G$ without $4$-cycles adjacent to $k$-cycles is…

A list assignment $L$ of a graph $G$ is a function that assigns to every vertex $v$ of $G$ a set $L(v)$ of colors. A proper coloring $\alpha$ of $G$ is called an $L$-coloring of $G$ if $\alpha(v)\in L(v)$ for every $v\in V(G)$. For a list…

For positive integers $a$ and $b$, a graph $G$ is $(a:b)$-choosable if, for each assignment of lists of $a$ colors to the vertices of $G,$ each vertex can be colored with a set of $b$ colors from its list so that adjacent vertices are…

DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-choosable.…

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 answer positively the question of Albertson asking whether every planar graph can be $5$-list-colored even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we…

Proper graph coloring assigns different colors to adjacent vertices of the graph. Usually, the number of colors is fixed or as small as possible. Consider applications (e.g. variants of scheduling) where colors represent limited resources…

A graph $G$ is {\em $k$-choosable} if for every assignment of a set $S(v)$ of $k$ colors to every vertex $v$ of $G$, there is a proper coloring of $G$ that assigns to each vertex $v$ a color from $S(v)$. We consider the complexity of…

There are two conjectures concerning planar graph colourings that are strengthenings of the four colour theorem. One concerns signed graph colouring and is proposed by M\'{a}\v{c}ajov\'{a}, Raspaud and \v{S}koviera. It asserts that every…