An upper bound on adaptable choosability of graphs

Given a (possibly improper) edge-colouring F of a graph G , a vertex-colouring c of G is adapted to F if no colour appears at the same time on an edge and on its two endpoints. If for some integer k , a graph G is such that given any list assignment L of G , with | L ( v ) | ≥ k for all v , and any edge-colouring F of G , there exists a vertex-colouring c of G adapted to F such that c ( v ) ∈ L ( v ) for all v , then G is said to be adaptably k -choosable. The smallest k such that G is adaptably k -choosable is called the adaptable choice number and is denoted by c h a d ( G ) . This note proves that c h a d ( G ) ≤ ⌈ M a d ( G ) / 2 ⌉ + 1 , where M a d ( G ) is the maximum of 2 | E ( H ) | / | V ( H ) | over all subgraphs H of G . As a consequence, we give bounds for classes of graphs embeddable into surfaces of non-negative Euler characteristics.