Acyclic Chromatic Index of Triangle-free 1-Planar Graphs

Abstract

An acyclic edge coloring of a graph G is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index χa′(G) of a graph G is the least number of colors in an acyclic edge coloring of G. It was conjectured that χa′(G)≤Δ(G)+2 for any simple graph G with maximum degree Δ(G). A graph is 1-planar if it can be drawn on the plane such that every edge is crossed by at most one other edge. In this paper, we show that every triangle-free 1-planar graph G has an acyclic edge coloring with Δ(G) + 16 colors.