On S-Packing Coloring of Subcubic Graphs

Given a sequence $S=(s_1,s_2,\dots ,s_k)$ of positive integers satisfying $s_1\le s_2\le \cdots \le s_k$, an $S$-packing coloring of a graph $G$ is a partition of $V(G)$ into $k$ subsets $V_1,V_2,\dots ,V_k$ such that, for each $1\le i\le k$, the distance between any two distinct vertices $x,y\in V_i$ is at least $s_i+1$. Yang and Wu established that every $3$-irregular subcubic graph admits a $(1,1,3)$-packing coloring. Later, Mortada and Togni introduced the concept of an $i$-saturated subcubic graph, defined as a subcubic graph in which every vertex of degree three has at most $i$ neighbors of degree three for $0\le i\le 3$. They further demonstrated that all $1$-saturated subcubic graphs are $(1,1,2)$-packing colorable. In this paper, we present new concise proofs of these results using a novel tool.