$S$-Packing Coloring of Cubic Halin Graphs

TL;DR

This paper investigates $S$-packing colorings of cubic Halin graphs, establishing their colorability under specific sequences, which advances understanding of graph coloring constraints in this class.

Contribution

The paper proves that all cubic Halin graphs are $(1,1,2,3)$- and $(1,2,2,2,2,2)$-packing colorable, providing new results on their coloring properties.

Findings

Cubic Halin graphs are $(1,1,2,3)$-packing colorable.

Cubic Halin graphs are $(1,2,2,2,2,2)$-packing colorable.

New bounds on $S$-packing colorings for cubic Halin graphs.

Abstract

Given a non-decreasing sequence $S = (s_{1}, s_{2}, \ldots , s_{k})$ of positive integers, an $S$-packing coloring of a graph $G$ is a partition of the vertex set of $G$ into $k$ subsets $\{V_{1}, V_{2}, \ldots , V_{k}\}$ such that for each $1 \leq i \leq k$, the distance between any two distinct vertices $u$ and $v$ in $V_{i}$ is at least $s_{i} + 1$. In this paper, we study the problem of $S$-packing coloring of cubic Halin graphs, and we prove that every cubic Halin graph is $(1,1,2,3)$-packing colorable. In addition, we prove that such graphs are $(1,2,2,2,2,2)$-packing colorable.