Acyclic Edge Coloring of 1-planar Graphs without 4-cycles

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles in G. The acyclic chromatic index \(\cal{X}_{\alpha}^{\prime}(G)\) of G is the smallest k such that G has an acyclic edge coloring using k colors. It was conjectured that every simple graph G with maximum degree Δ has \(\cal{X}_{\alpha}^{\prime}(G)\le\Delta+2\). A 1-planar graph is a graph that can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, we show that every 1-planar graph G without 4-cycles has \(\cal{X}_{\alpha}^{\prime}(G)\le\Delta+22\).