Packing chromatic number of subcubic graphs

TL;DR

This paper demonstrates that the packing chromatic number of subcubic graphs is unbounded, providing constructions and probabilistic results that show large packing chromatic numbers in such graphs.

Contribution

It proves that the packing chromatic number of subcubic graphs is unbounded and shows that most large cubic graphs with high girth have arbitrarily large packing chromatic numbers.

Findings

Existence of subcubic graphs with arbitrarily large packing chromatic number

Most large cubic graphs with girth at least g have packing chromatic number greater than k

Answer to an open question about boundedness of packing chromatic number in subcubic graphs

Abstract

A packing $k$-coloring of a graph $G$ 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 $i+1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. Sloper showed that there are $4$-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed $k$ and $g\geq 2k+2$, almost every $n$-vertex cubic graph of girth at least $g$ has the packing chromatic number greater than $k$.