Generalized Chromatic Numbers of Random Graphs

Abstract

Given a hereditary (closed under taking induced subgraphs) class of graphs $\mathcal{P}$, the $\mathcal{P}$-chromatic number of a graph G, denoted $\chi _\mathcal{P} ( G )$, is defined to be the least integer k such that $V( G )$ admits a partition into k subsets each of which induce a member of $\mathcal{P}$. The $\mathcal{P}$-chromatic number of random graphs G on n vertices with fixed edge probability $0 < p < 1$ is studied and it is shown that $\chi _\mathcal{P} ( G ) \sim cn$ in case $\mathcal{P} < \infty $ and $\chi _\mathcal{P} ( G ) = \Theta ( n/\log n )$ in case $| \mathcal{P} | = \infty $. Also considered are generalized edge chromatic numbers.