A note on the acyclic 3-choosability of some planar graphs

Hervé Hocquard (1) , Mickael Montassier (1) , André Raspaud (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

An acyclic coloring of a graph $G$ is a coloring of its vertices such that : (i) no two adjacent vertices in $G$ receive the same color and (ii) no bicolored cycles exist in $G$. A list assignment of $G$ is a function $L$ that assigns to each vertex $v\,\in\,V(G)$ a list $L(v)$ of available colors. Let $G$ be a graph and $L$ be a list assignment of $G$. The graph $G$ is acyclically $L$-list colorable if there exists an acyclic coloring $\phi$ of $G$ such that $\phi(v)\,\in\,L(v)$ for all $v\,\in\,V(G).$ If $G$ is acyclically $L$-list colorable for any list assignment $L$ with $|L(v)|\,\ge\,k$ for all $v\,\in\,V(G)$, then $G$ is acyclically $k$-choosable. In this paper, we prove that every planar graph with neither cycles of lengths 4 to 7 (resp. to 8, to 9, to 10) nor triangles at distance less 7 (resp. 5, 3, 2) is acyclically 3-choosable.

Domaines: Mathématique discrète [cs.DM]

Hervé Hocquard, Mickael Montassier, André Raspaud. A note on the acyclic 3-choosability of some planar graphs. 2009. ⟨hal-00425986⟩