On the inclusion chromatic index of a Halin graph

An inclusion-free edge-coloring of a graph G with $\delta \left(G\right)\ge 2$ is a proper edge-coloring such that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. The minimum number of colors needed in an inclusion-free edge-coloring of G is called the $inclusion$-$free$ $chromaticindex$, denoted by ${\chi }_{\subset }^{\prime }\left(G\right)$. In this paper, we show that for a Halin graph G with maximum degree $\Delta \ge 4$, if G is isomorphic to a wheel $W_{\Delta +1}$ where Δ is odd, then ${\chi }_{\subset }^{\prime }\left(G\right)=\Delta +2$, otherwise ${\chi }_{\subset }^{\prime }\left(G\right)=\Delta +1$. We also show a special cubic Halin graph with ${\chi }_{\subset }^{\prime }\left(G\right)=5$.