On the strong chromatic index of cubic 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 they are incident to a common edge or share an endpoint. 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 without vertices of degree two by connecting all leaves through a cycle. If a cubic Halin graph GG is different from two particular graphs Ne2Ne2 and Ne4Ne4, then we prove sχ′(G)⩽7sχ′(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.