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 Xα′(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{X}_{\alpha}^{\prime}(G)$$\end{document} 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 Xα′(G)≤Δ+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{X}_{\alpha}^{\prime}(G)\le\Delta+2$$\end{document}. 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 Xα′(G)≤Δ+22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{X}_{\alpha}^{\prime}(G)\le\Delta+22$$\end{document}.