An upper bound on the adaptable choosability of graphs.

Mickael Montassier (1) , André Raspaud (1) , Xuding Zhu (2)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 SYSU - Sun Yat-sen University [Guangzhou]

Given a (possibly improper) edge-colouring $F$ of a graph $G$, a vertex colouring $c$ of $G$ is \emph{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 $|L(v)| \ge k$ for all $v$, and any edge-colouring $F$ of $G$, there exists a vertex colouring $c$ of $G$ adapted to $F$ such that $c(v) \in L(v)$ for all $v$, then $G$ is said to be \emph{adaptably $k$-choosable}. The smallest $k$ such that $G$ is adaptably $k$-choosable is called the \emph{adaptable choice number} and is denoted by $ch_{ad}(G)$. This note proves that $ch_{ad}(G) \le \lceil {Mad}(G)/2 \rceil +1$, where ${Mad}(G)$ 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.

Domaines: Mathématique discrète [cs.DM]

Mickael Montassier, André Raspaud, Xuding Zhu. An upper bound on the adaptable choosability of graphs.. European Journal of Combinatorics, 2009, 30 (2), pp.351-355. ⟨10.1016/j.ejc.2008.06.003⟩. ⟨hal-00322868⟩