The adaptable choosability number of a multigraph G, denoted cha(G), is the smallest integer k such that every edge labeling of G and assignment of lists of size k to the vertices of G permits a list coloring of G in which no edge e = uv has both u and v colored with the label of e. We show that cha grows with ch, i.e. there is a function f tending to infinity such that cha(G) ≥ f(ch(G)).

We answer two questions of Zhu on circular choosability of graphs. We show that the circular list chromatic number of an even cycle is equal to 2 and give an example of a graph for which the infimum in the definition of the circular list chromatic number is not attained.

In this paper, we prove a structural theorem of Lebesgue’s type concerning some unavoidable con1gurations for graphs which can be embedded on surfaces of nonnegative characteristic and in which no two 3-cycles share a common vertex. As a corollary, we get a result about choosability of graphs embedded in surface of positive characteristic. c © 2002 Elsevier Science B.V. All rights reserved.