An upper bound for the semistrong chromatic index of Halin graphs

For a graph $G$, a semistrong matching is a matching $M$ such that every edge in $M$ contains at least one endpoint of degree one in the induced subgraph $G[V(M)]$. The semistrong chromatic index ${\chi }_{ss}^{\mathrm{\prime }}(G)$ denotes the minimum number of colors required for a proper edge-coloring where each color class induces a semistrong matching. We study this parameter for Halin graphs, which are planar graphs formed by connecting all leaves of a tree $T$ (with no degree-two vertices) via an outer cycle $C$. Our main result establishes that for any Halin graph $G=T\cup C$ with maximum degree $\mathrm{\Delta }(G)$, the semistrong chromatic index satisfies ${\chi }_{ss}^{\mathrm{\prime }}(G)\le \mathrm{\Delta }(G)+4$, with equality attained by the wheel graphs $W_4$ and $W_7$.