List colorings of K5-minor-free graphs with special list assignments

Abstract

The following question was raised by Bruce Richter. Let G be a planar, 3-connected graph that is not a complete graph. Denoting by d(v) the degree of vertex v, is G L-list colorable for every list assignment L with |L(v)| = min{d(v), 6} for all v∈V(G)? More generally, we ask for which pairs (r, k) the following question has an affirmative answer. Let r and k be the integers and let G be a K5-minor-free r-connected graph that is not a Gallai tree (i.e. at least one block of G is neither a complete graph nor an odd cycle). Is G L-list colorable for every list assignment L with |L(v)| = min{d(v), k} for all v∈V(G)? We investigate this question by considering the components of G[Sk], where Sk: = {v∈V(G)|d(v)8k} is the set of vertices with small degree in G. We are especially interested in the minimum distance d(Sk) in G between the components of G[S k]. © 2011 Wiley Periodicals, Inc.