On S-Packing Coloring of Subcubic Graphs
Hadeel Al Bazzal

TL;DR
This paper explores S-packing colorings of subcubic graphs, providing simplified proofs for existing results and introducing a new tool to analyze colorability based on vertex degree saturation.
Contribution
The authors offer concise proofs of known coloring results for subcubic graphs and introduce a novel analytical tool for studying S-packing colorings.
Findings
Every 3-irregular subcubic graph admits a (1,1,3)-packing coloring.
All 1-saturated subcubic graphs are (1,1,2)-packing colorable.
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 -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 -saturated subcubic graphs are \( (1,1,2) \)-packing colorable. In this paper, we present new concise proofs of these results using a novel tool.
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Taxonomy
TopicsGraph Labeling and Dimension Problems · graph theory and CDMA systems · Advanced Graph Theory Research
