Packing chromatic vertexcritical graphs
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 vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertexcritical graphs, $\chi_{\rho}$critical for short, are introduced as the graphs $G$ for which $\chi_{\rho}(Gx) < \chi_{\rho}(G)$ holds for every vertex $x$ of $G$. If $\chi_{\rho}(G) = k$, then $G$ is $k$$\chi_{\rho}$critical. It is shown that if $G$ is $\chi_{\rho}$critical, then the set $\{\chi_{\rho}(G)  \chi_{\rho}(Gx):\ x\in V(G)\}$ can be almost arbitrary. The $3$$\chi_{\rho}$critical graphs are characterized, and $4$$\chi_{\rho}$critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$$\chi_{\rho}$critical tree and that a $k$$\chi_{\rho}$critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertextransitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G)$, then $G\,\square\, H$ is $\chi_{\rho}$critical.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 DOI:
 10.48550/arXiv.1810.03904
 arXiv:
 arXiv:1810.03904
 Bibcode:
 2018arXiv181003904K
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 Discrete Mathematics &