Characterization of $3$-Regular Halin Graphs with Edge-face Total Chromatic Number Equal to Four

Abstract

The edge-face total chromatic number of $3$-regular Halin graphs was shown to be $4$ or $5$ in $[5]$. In this paper, we shall provide a necessary and sufficient condition to characterize $3$-regular Halin graphs with edge-face total chromatic number equal to four.