Packing $(1,1,2,4)$-coloring of subcubic outerplanar graphs

TL;DR

This paper investigates packing colorings of subcubic outerplanar graphs, establishing specific colorability bounds and demonstrating the sharpness of these bounds through counterexamples.

Contribution

It proves that every subcubic outerplanar graph can be colored with a (1,1,2,4) packing coloring and identifies the limits of such colorings.

Findings

Every 2-connected subcubic outerplanar graph has a (1,1,2)-coloring.

All subcubic outerplanar graphs are (1,1,2,4)-colorable.

Counterexamples show the bounds are tight and cannot be improved to certain other colorings.

Abstract

For $1\leq s_1 \le s_2 \le \ldots \le s_k$ and a graph $G$, a packing $(s_1, s_2, \ldots, s_k)$-coloring of $G$ is a partition of $V(G)$ into sets $V_1, V_2, \ldots, V_k$ such that, for each $1\leq i \leq k$, the distance between any two distinct $x,y\in V_i$ is at least $s_i + 1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the smallest $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring. It is known that there are trees of maximum degree 4 and subcubic graphs $G$ with arbitrarily large $\chi_p(G)$. Recently, there was a series of papers on packing $(s_1, s_2, \ldots, s_k)$-colorings of subcubic graphs in various classes. We show that every $2$-connected subcubic outerplanar graph has a packing $(1,1,2)$-coloring and every subcubic outerplanar graph is packing $(1,1,2,4)$-colorable. Our results are sharp in the sense that there are $2$-connected subcubic outerplanar graphs that are not packing $(1,1,3)$-colorable and there are subcubic outerplanar graphs that are not packing $(1,1,2,5)$-colorable. We also show subcubic outerplanar graphs that are not packing $(1,2,2,4)$-colorable and not packing $(1,1,3,4)$-colorable.