An improved upper bound on the edge-face coloring of 2-connected plane graphs

Abstract

The edge-face chromatic number ${\chi }_{ef}\left(G\right)$ of a plane graph G is the least number of colors such that any two adjacent or incident elements in $E\left(G\right)\cup F\left(G\right)$ receive different colors. In 2005, Luo and Zhang proved that each 2-connected simple graph G with $\Delta \ge 24$ has ${\chi }_{ef}\left(G\right)=\Delta$. The condition $\Delta ⩾24$ is improved to $\Delta ⩾13$ in this paper.