The strong chromatic index of Halin graphs

Abstract

A strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if their distance is at most two. The strong chromatic index of a graph G, denoted by sχ′(G), is the minimum number of colors needed for a strong edge coloring of G. A Halin graph G is a plane graph constructed from a tree T without vertices of degree two by connecting all leaves through a cycle C. If a Halin graph G=T∪C is different from a certain necklace Ne2 and any wheel Wn, n≢0(mod3), then we prove that sχ′(G)⩽sχ′(T)+3.