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 $\mathrm{\Delta }\ge 9$ is of class one. Meanwhile, we show that there exist class two $1$-planar graphs with maximum degree $\mathrm{\Delta }$ for each $\mathrm{\Delta }\le 7$.