On (1,1,2,3)- and (1,1,3,3,3)-Packing Colorings of Claw-Free Subcubic Graphs

Abstract

For a non-decreasing sequence $S=(a_1,a_2,\dots ,a_r)$ of positive integers, an $S$-packing coloring of a graph $G$ is a partition of $V(G)$ into sets $A_1,\dots ,A_r$ such that any two distinct vertices in $A_i$ are at distance greater than $a_i$, for every $i\in \{1,\dots ,r\}$. Gastineau and Togni [\emph{Discrete Math.} 339 (2016), 2461--2470] asked whether every subcubic graph, except the Petersen graph, is $(1,1,2,3)$-packing colorable. In this paper, we prove that every claw-free subcubic graph is $(1,1,2,3)$-packing colorable. Moreover, we show that every connected claw-free subcubic graph, except a single graph $\mathcal{H}$, is $(1,1,3,3,3)$-packing colorable, thereby confirming a conjecture of the first two authors. Both results are best possible. Our proofs rely on a structural framework based on the skeleton and core graphs of a claw-free subcubic graph, together with a Hall-type matching argument that reduces the construction of suitable $3$-packings to a matching problem in an auxiliary bipartite graph.