We consider the following type of problems. Given a graph G = (V; E) and lists L(v) of allowed colors for its vertices v 2 V such that jL(v)j = p for all v 2 V and jL(u) \ L(v)j c for all uv 2 E, is it possible to nd a \list coloring", i.e., a color f (v) 2 L(v) for each v 2 V , so that f (u) 6 = f (v) for all uv 2 E ? We prove that every graph of maximum degree admits a list coloring for every...

It is proved that, if G is a K4-minor-free graph with maximum degree 3, then G is totally 4-choosable; that is, if every element (vertex or edge) of G is assigned a list of 4 colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that...

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...

Given a group A and a directed graph G, let F(G, A) denote the set of all maps f : E(G) → A. Fix an orientation of G and a list assignment L : V (G) → 2A. For an f ∈ F(G, A), G is (A, L , f )-colorable if there exists a map c : V (G) → ∪v∈V (G)L(v) such that c(v) ∈ L(v), ∀v ∈ V (G) and c(x)− c(y) = f (xy) for every edge e = xy directed from x to y. If for any f ∈ F(G, A), G has an (A, L , f )-c...

A proper vertex coloring of a graph G = (V , E) is acyclic if G contains no bicolored cycle. Given a list assignment L = {L(v) | v ∈ V } of G, we say G is acyclically L-list colorable if there exists a proper acyclic coloring π of G such that π(v) ∈ L(v) for all v ∈ V . If G is acyclically L-list colorable for any list assignment with |L(v)| ≥ k for all v ∈ V , then G is acyclically k-choosable...

In this paper, we study the group and list group colorings of total graphs and present group coloring versions of the total and list total colorings conjectures.We establish the group coloring version of the total coloring conjecture for the following classes of graphs: graphs with small maximum degree, two-degenerate graphs, planner graphs with maximum degree at least 11, planner graphs withou...

A graph G = (V ,E) is list L-colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is list L-colorable for every list assignment with |L(v)| k for all v ∈ V , then G is said to be k-choosable. In this paper, we prove that (1) every planar graph either without 4and 5-cycles, and without triangles at distance...

A plane graph G is said to be k-edge-face choosable if, for every list L of colors satisfying |L(x)| = k for every edge and face x , there exists a coloring which assigns to each edge and each face a color from its list so that any adjacent or incident elements receive different colors. We prove that every plane graph G with maximum degree ∆(G) is (∆(G)+ 3)-edge-face choosable. © 2004 Elsevier ...

For each proper subgraph H of K5, we determine all pairs (k, d) such that every H-minor-free graph is (k, d)-choosable or (k, d)-choosable. The main structural lemma is that the only 3-connected (K5 − e)-minor-free graphs are wheels, the triangular prism, and K3,3; this is used to prove that every (K5 − e)-minor-free graph is 4-choosable and (3, 1)-choosable.

It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ǫ, and k and d are positive integers such that k > 3 and d is sufficiently large in terms of k and ǫ, then G is (k, d)∗-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the kno...