Packing chromatic number under local changes in a graph

TL;DR

This paper studies the packing chromatic number in graphs, establishing a lower bound for subcubic graphs and analyzing how local modifications like edge subdivision affect this number.

Contribution

It proves a new lower bound for the packing chromatic number in subcubic graphs and examines its behavior under local graph operations such as edge subdivision.

Findings

Packing chromatic number exceeds 13 in subcubic graphs.

Edge subdivision changes the packing chromatic number within a specific range.

Provides bounds for the packing chromatic number after local modifications.

Abstract

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. It is proved that in the class of subcubic graphs the packing chromatic number is bigger than $13$, thus answering an open problem from [Gastineau, Togni, $S$-packing colorings of cubic graphs, Discrete Math.\ 339 (2016) 2461--2470]. In addition, the packing chromatic number is investigated with respect to several local operations. In particular, if $S_e(G)$ is the graph obtained from a graph $G$ by subdividing its edge $e$, then $\left\lfloor \chi_{\rho}(G)/2 \right\rfloor +1 \le \chi_{\rho}(S_e(G)) \le \chi_{\rho}(G)+1$.