Packing chromatic number versus chromatic and clique number

TL;DR

This paper explores the relationships between the packing chromatic number, chromatic number, and clique number of graphs, establishing realizability conditions and bounds, with specific focus on Mycielskian graphs.

Contribution

It characterizes when certain triples of graph invariants are realizable and provides bounds on the packing chromatic number using graph parameters.

Findings

Proves that if chromatic number equals packing chromatic number and both are at least 3, then the clique number equals the chromatic number.

Shows that triples (2, k, k+1) and (2, k, k+2) are not realizable for k ≥ 4.

Provides bounds on the packing chromatic number based on maximum degree and independence number.

Abstract

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where each $V_i$ is an $i$-packing. In this paper, we investigate for a given triple $(a,b,c)$ of positive integers whether there exists a graph $G$ such that $\omega(G) = a$, $\chi(G) = b$, and $\chi_{\rho}(G) = c$. If so, we say that $(a, b, c)$ is realizable. It is proved that $b=c\ge 3$ implies $a=b$, and that triples $(2,k,k+1)$ and $(2,k,k+2)$ are not realizable as soon as $k\ge 4$. Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on $\chi_{\rho}(G)$ in terms of $\Delta(G)$ and $\alpha(G)$ is also proved.