Fast algorithm for $S$-packing coloring of Halin graphs
Xin Zhang, Dezhi Zou
TL;DR
This work addresses the S-packing coloring problem on Halin graphs with maximum degree $Δ≤5$ by presenting a linear-time algorithm that constructs a $(1^2,2^3)$-packing coloring. The method combines a two-coloring of the characteristic tree with a recoloring of the adjoint cycle and a subsequent conflict-resolution pass using colors from $\\{1,1',2_a,2_b,2_c\\}$, ensuring all distance constraints are satisfied. The authors prove correctness via a sequence of structural lemmas and show the overall complexity is $O(|G|)$, establishing the first linear-time $(1^2,2^3)$-packing coloring for this graph class and highlighting that some Halin graphs are not $(1,2,2,2)$-packing colorable. The results advance the understanding of S-packing colorings in Halin graphs and have potential implications for related frequency assignment problems and graph subdivision bounds in subcubic contexts.
Abstract
Motivated by frequency assignment problems in wireless broadcast networks, Goddard, Hedetniemi, Hedetniemi, Harris, and Rall introduced the notion of $S$-packing coloring in 2008. 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 its vertex set 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 $u, v \in V_i$ is at least $s_i + 1$. In this paper, we study the $S$-packing coloring problem for Halin graphs with maximum degree $Δ\leq 5$. Specifically, we present a linear-time algorithm that constructs a $(1,1,2,2,2)$-packing coloring for any Halin graph satisfying $Δ\leq 5$. It is worth noting that there are Halin graphs that are not $(1,2,2,2)$-packing colorable.