Acyclic L-coloring of graphs with maximum degrees 5 and 6

Abstract For a graph G = ( V ( G ) , E ( G ) ) , an acyclic coloring of G is a proper vertex coloring such that every cycle is colored with at least three colors. A graph G is acyclically L -colorable if for a given list assignment L = { L ( v ) | v ∈ V ( G ) } , there exists an acyclic coloring f such that f ( v ) ∈ L ( v ) for all v ∈ V ( G ) . If G is acyclically L -colorable for every list assignment L with | L ( v ) | ≥ k for all v ∈ V ( G ) , then G is acyclically k -choosable. In this paper, we show that every graph with maximum degree 5 is acyclically 7-choosable and every graph with maximum degree 6 is acyclically 10-choosable.