On the packing chromatic number of Moore graphs

TL;DR

This paper investigates the packing chromatic number of Moore graphs with girths 6, 8, and 12, providing exact values for some cases and bounds for others, advancing understanding of graph coloring in highly regular structures.

Contribution

It determines the packing chromatic number for (q+1,6)-Moore graphs and relates it to geometric structures for girth 8, while establishing bounds for girth 12 graphs.

Findings

Exact packing chromatic number for girth 6 Moore graphs.

Determination of the number in terms of generalized quadrangles for girth 8.

Bounds for the packing chromatic number when girth is 12 and q ≥ 9.

Abstract

The \emph{packing chromatic number $\chi_\rho (G)$} of a graph $G$ is the smallest integer $k$ for which there exists a vertex coloring $\Gamma: V(G)\rightarrow \{1,2,\dots , k\}$ such that any two vertices of color $i$ are at distance at least $i + 1$. For $g\in \{6,8,12\}$, $(q+1,g)$-Moore graphs are $(q+1)$-regular graphs with girth $g$ which are the incidence graphs of a symmetric generalized $g/2$-gons of order $q$. In this paper we study the packing chromatic number of a $(q+1,g)$-Moore graph $G$. For $g=6$ we present the exact value of $\chi_\rho (G)$. For $g=8$, we determine $\chi_\rho (G)$ in terms of the intersection of certain structures in generalized quadrangles. For $g=12$, we present lower and upper bounds for this invariant when $q\ge 9$ an odd prime power.