An upper bound on adaptable choosability of graphs

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