On S-Packing Coloring of Subcubic Graphs

Given a sequence ( S = (s_1, s_2, \ldots, s_k) ) of positive integers satisfying ( s_1 \leq s_2 \leq \dots \leq 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 \leq i \leq 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 \leq i \leq 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.

💡 Research Summary

The paper revisits two known results concerning S‑packing colorings of subcubic graphs: every 0‑saturated (i.e., 3‑irregular) subcubic graph admits a (1,1,3)‑packing coloring, and every 1‑saturated subcubic graph admits a (1,1,2)‑packing coloring. While these facts were previously proved by Yang and Wu, and by Mortada and Togni through intricate case analyses, the authors present a unified and much shorter approach based on a single structural tool.

The core of the method is the selection of a spanning bipartite subgraph B = X ∪ Y of the given graph G that contains the maximum possible number of edges. Proposition 2.1 shows that for any vertex v, the degree in B satisfies d_B(v) ≥ ½ d_G(v). When G is subcubic, this immediately implies (Corollary 2.1) that the induced subgraphs G