Acyclic Chromatic Index of 1-Planar Graphs

The acyclic chromatic index χa′(G) of a graph G is the smallest k for which G is a proper edge colorable using k colors. A 1-planar graph is a graph that can be drawn in plane such that every edge is crossed by at most one other edge. In this paper, we prove that every 1-planar graph G has χa′(G)≤Δ+36, where Δ denotes the maximum degree of G. This strengthens a result that if G is a triangle-free 1-planar graph, then χa′(G)≤Δ+16.