Acyclic chromatic index of triangle-free 1-planar graphs

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) ≤ ∆(G) + 2 for any simple graph G with maximum degree ∆(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 prove that every triangle-free 1-planar graph G admits an acyclic edge coloring with ∆(G) + 17 colors.