Proof of a conjecture on 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 graph G, denoted by χs′(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. In 2018, Hu et al. (2018) posed a conjecture which says that if G=T∪C is a Halin graph other than a wheel Wn, Ne2, or Ne4, then χs′(G)≤χs′(T)+2. The bound can be achieved. In this paper, we show that the above conjecture is true. This improves the known bound χs′(G)≤χs′(T)+3 obtained by Lai et al. (2012), and extends the results on Halin graphs G with Δ(G)≤3 obtained by Lih and Liu (2012) and Δ(G)≤4 obtained by Hu et al. (2018), respectively.