A SUFFICIENT CONDITION FOR ACYCLIC 5-CHOOSABILITY OF PLANAR GRAPHS WITHOUT 5-CYCLES

A proper vertex coloring of a graph G is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment $L=\{L(v):v{\in}V(G)\}$, there exists an acyclic coloring ${\phi}$ of G such that ${\phi}(v){\in}L(v)$ for all $v{\in}V(G)$ A graph G is acyclically k-choosable if G is acyclically L-list colorable for any list assignment with $L(v){\geq}k$ for all $v{\in}V(G)$. Let G be a planar graph without 5-cycles and adjacent 4-cycles. In this article, we prove that G is acyclically 5-choosable if every vertex v in G is incident with at most one i-cycle, $i {\in}\{6,7\}$.