On Edge Colorings of \(1\)-Planar Graphs without Chordal \(5\)-Cycles

Abstract

A graph is \(1\)-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that every \(1\)-planar graph without chordal \(5\)-cycles and with maximum degree \(\Delta \geq 9\) is of class one. Meanwhile, we show that there exist class two \(1\)-planar graphs with maximum degree \(\Delta\) for each \(\Delta \leq 7\).