Thomassen proved that any plane graph of girth 5 is list-colorable from any list assignment such that all vertices have lists of size two or three and the vertices with list of size two are all incident with the outer face and form an independent set. We present a strengthening of this result, relaxing the constraint on the vertices with list of size two. This result is used to bound the size o...

A graph is `-choosable if, for any choice of lists of ` colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs when the number k of colors is limited. We get results which differ surprisingly from the usual case where k is implicit and which extend known results for the usual ca...

It is proved that if G is a plane embedding of a K4-minor-free graph with maximum degree ∆, then G is entirely 7-choosable if ∆ ≤ 4 and G is entirely (∆+2)-choosable if ∆ ≥ 5; that is, if every vertex, edge and face of G is given a list of max{7,∆+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is pr...

A k-list assignment L of a graph G is a mapping which assigns to each vertex v of G a set L(v) of size k. A (k,t)-list assignment of G is a k-list assignment with | ⋃ v∈V (G) L(v)| = t. An L-coloring φ of G is a proper coloring of G such that φ(v) is chosen from L(v) for every vertex v. A graph G is Lcolorable if G has an L-coloring. When the parameter t is not of special interest, we simply sa...

A graph G is 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 deciding whether a given graph is k-choosable for some constant k. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem o...

A solution to a problem of Erdős, Rubin and Taylor is obtained by showing that if a graph G is (a : b)-choosable, and c/d > a/b, then G is not necessarily (c : d)-choosable. The simplest case of another problem, stated by the same authors, is settled, proving that every 2-choosable graph is also (4 : 2)-choosable. Applying probabilistic methods, an upper bound for the k choice number of a graph...

There exists a variety of coloring problems for plane graphs, involving vertices, edges, and faces in all possible combinations. For instance, the entire graph we are to color these three sets so that any pair adjacent or incident elements get different colors. We study here some this type from algebraic perspective, focusing on facial variant. obtain several results concerning Alon–Tarsi numbe...