On the strong chromatic index of cubic Halin graphs

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 they are incident to a common edge or share an endpoint. The strong chromatic index of a graph G , denoted by s χ ′ ( 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 without vertices of degree two by connecting all leaves through a cycle. If a cubic Halin graph G is different from two particular graphs N e 2 and N e 4 , then we prove s χ ′ ( G ) ⩽ 7 . This solves a conjecture proposed in W.C. Shiu, W.K. Tam, The strong chromatic index of complete cubic Halin graphs, Appl. Math. Lett. 22 (2009) 754–758.