A diamond-free claw-free cubic graph with strong chromatic index 7

Abstract

A strong edge coloring is a proper edge coloring in which every color class is an induced matching; the least number of colors is the strong chromatic index $χ'_s(G)$. Lin and Lin proved that every claw-free subcubic graph other than the triangular prism satisfies $χ'_s(G) \le 7$, with all their tight examples containing diamonds. Kardos (Problem 4.1 of the open-problem collection of the 33rd Workshop on Cycles and Colourings) asked whether every diamond-free claw-free cubic graph is strongly 6-edge-colorable, equivalently whether $χ'_s(T(G))=6$ for every cubic graph $G$, where $T(G)$ is the truncation of $G$. We exhibit an explicit connected, simple, diamond-free, claw-free cubic graph $H$ on 18 vertices with $χ'_s(H)=7$, and show that it has the fewest vertices possible for such a non-prism example. Hence the first formulation, over simple cubic graphs, is false even after excluding the prism; and since $H$ is the truncation of a cubic multigraph with parallel edges, the two formulations are not equivalent unless "cubic graph" is allowed to mean loopless multigraph, under which reading the problem is answered negatively. The narrower version restricted to truncations of simple base graphs remains open.