Fast algorithm for $S$-packing coloring of Halin graphs

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 $\Delta \leq 5$. Specifically, we present a linear-time algorithm that constructs a $(1,1,2,2,2)$-packing coloring for any Halin graph satisfying $\Delta \leq 5$. It is worth noting that there are Halin graphs that are not $(1,2,2,2)$-packing colorable.