Packing colorings of subcubic outerplanar graphs

TL;DR

This paper investigates the packing chromatic number of subcubic outerplanar graphs, establishing bounds and coloring properties for various subclasses, revealing unboundedness in general but boundedness in specific cases.

Contribution

It proves that 2-connected bipartite subcubic outerplanar graphs have a packing chromatic number at most 7 and characterizes coloring possibilities for triangle-free and bipartite subclasses.

Findings

The packing chromatic number is bounded by 7 for 2-connected bipartite subcubic outerplanar graphs.

Certain subcubic outerplanar graphs with triangles cannot be colored with (1,2,2,2) or (1,2,2,3) sequences.

Bipartite outerplanar graphs admit an S-packing coloring with S=(1,3,...,3), but not with S where 3 is replaced by 4.

Abstract

Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a $(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an $S$-packing coloring for $S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the maximum degree of vertices), and this property does not hold if one of the integers $3$ is replaced by $4$ in the sequence $S$.