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

TL;DR

This paper proves that subcubic graphs with a maximum average degree less than 30/11 are packing (1,1,2,2)-colorable, supporting a conjecture about the packing chromatic number of subdivided subcubic graphs.

Contribution

It establishes a new bound on maximum average degree ensuring packing (1,1,2,2)-colorability in subcubic graphs, advancing understanding of the packing chromatic number.

Findings

Subcubic graphs with mad(G)<30/11 are packing (1,1,2,2)-colorable.

Supports the conjecture that all subdivided subcubic graphs have packing chromatic number at most 5.

Provides a sufficient condition related to maximum average degree for packing colorability.

Abstract

For a sequence of non-decreasing positive integers $S = (s_1, \ldots, s_k)$, a packing $S$-coloring is a partition of $V(G)$ into sets $V_1, \ldots, V_k$ 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 smallest $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring is called the packing chromatic number of $G$ and is denoted by $\chi_p(G)$. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The question whether $\chi_p(D(G)) \le 5$ for all subcubic graphs was first asked by Gastineau and Togni and later conjectured by Bresar, Klavzar, Rall and Wash. Gastineau and Togni observed that if one can prove every subcubic graph except the Petersen graph is packing $(1,1,2,2)$-colorable then the conjecture holds. The maximum average degree, mad($G$), is defined to be $\max\{\frac{2|E(H)|}{|V(H)|}: H \subset G\}$. In this paper, we prove that subcubic graphs with $mad(G)<\frac{30}{11}$ are packing $(1,1,2,2)$-colorable. As a corollary, the conjecture of Bresar et al holds for every subcubic graph $G$ with $mad(G)<\frac{30}{11}$.