On S-Packing Coloring of Subcubic Graphs

Abstract

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.