The strong chromatic index of Halin graphs

A strong edge coloring   of a graph GG is an assignment of colors to the edges of GG such that two distinct edges are colored differently if their distance is at most two. The strong chromatic index   of a graph GG, denoted by sχ′(G)sχ′(G), is the minimum number of colors needed for a strong edge coloring of GG. A Halin graph GG is a plane graph constructed from a tree TT without vertices of degree two by connecting all leaves through a cycle CC. If a Halin graph G=T∪CG=T∪C is different from a certain necklace Ne2Ne2 and any wheel WnWn, n≢0(mod3), then we prove that sχ′(G)⩽sχ′(T)+3sχ′(G)⩽sχ′(T)+3.