On S-Packing Coloring of Subcubic Graphs

Given a sequence S = (s₁, s₂, …, sₖ) of positive integers with s₁ ≤ s₂ ≤ … ≤ sₖ, an S-packing coloring of a graph G is a partition of $V(G)$ into k subsets V₁, V₂, …, Vₖ such that for each 1 ≤ i ≤ k the distance between any two distinct x, y ∈ Vᵢ is at least sᵢ + 1. In 2023, Yang and Wu proved that all 3-irregular subcubic graphs are $(1,1,3)$-packing colorable. In 2024, Mortada and Togni proved that every 1-saturated subcubic graph is $(1, 1, 2)$-packing colorable. In this paper, we provide new, concise proofs for these two theorems using a novel tool.