The star edge coloring of cubic Halin graphs with star chromatic index $5$

The star chromatic index of a graph $G$, denoted by ${\chi }_{st}^{\mathrm{\prime }}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. Casselgren et al. and Hou et al. independently proved that the star chromatic index of a cubic Halin graph, except a special graph, is at most $6$. It remains an open problem to determine which of such graphs have star chromatic index $5$. In this paper, we show that if $G\mathrm{\ne }{N}_{e_2}$ is a cubic Halin graph whose tree is a caterpillar or a complete tree, then ${\chi }_{st}^{\mathrm{\prime }}(G)=5$.