Edge-face coloring of 2-connected plane graphs

Abstract

The edge-face chromatic number $\chi_{\rm~ef}(G)$ of a plane graph $G$is the least number of colors such that any two adjacent edges,adjacent faces, and incident edge and face have different colors.In this paper, we show that if $G$ is a 2-connected simple plane graph with maximum degree$\Delta\ge~16$, then $\chi_{\rm~ef}(G)=\Delta$.This improves a known result that if $G$ is a 2-connected simple plane graph with $\Delta\ge~24$, then$\chi_{\rm~ef}(G)=\Delta$.