Some results on acyclic edge coloring of plane graphs

Abstract

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. The acyclic chromatic index of a graph G, denoted by α ′ ( G ) , is the minimum number k such that G admits an acyclic edge coloring using k colors. Let G be a plane graph with maximum degree Δ and girth g. In this paper, we prove that α ′ ( G ) = Δ ( G ) if one of the following conditions holds: (1) Δ ⩾ 8 and g ⩾ 7 ; (2) Δ ⩾ 6 and g ⩾ 8 ; (3) Δ ⩾ 5 and g ⩾ 9 ; (4) Δ ⩾ 4 and g ⩾ 10 ; (5) Δ ⩾ 3 and g ⩾ 14 . We also improve slightly a result of A. Fiedorowicz et al. (2008) [7] by showing that every triangle-free plane graph admits an acyclic edge coloring using at most Δ ( G ) + 5 colors.