Diamond-free, claw-free cubic graphs are (1, 1, 2, 3)-packing colorable

Abstract

A $(1,1,2,k)$-packing coloring of a graph $G$ is a partition of $V(G)$ into two independent sets, a 2-packing, and a $k$-packing. Recently, the question was posed in [A short proof that every claw-free cubic graph is (1, 1, 2, 2)-packing colorable, arXiv:2512.24001v1] as to whether every claw-free cubic graph is $(1,1,2,3)$-packing colorable. We provide an answer in the affirmative in the case that $G$ is a diamond-free, claw-free cubic graph.