Edge Coloring of \(1\)-Planar Graphs without Intersecting Triangles and Chordal \(5\)-Cycles

Abstract

A graph \(G\) is \(1\)-planar if it can be embedded in the plane \(\mathbb{R}^2\) so that each edge of \(G\) is crossed by at most one other edge. In this paper, we show that each \(1\)-planar graph of maximum degree \(\Delta\) at least \(7\) with neither intersecting triangles nor chordal \(5\)-cycles admits a proper edge coloring with \(\Delta\) colors.