Distinguishing Chromatic Number of Random Cayley graphs

TL;DR

This paper studies the distinguishing chromatic number of random Cayley graphs over abelian groups, showing that with high probability it is at most one more than the usual chromatic number.

Contribution

It establishes a probabilistic bound on the distinguishing chromatic number of random Cayley graphs, linking it closely to the standard chromatic number.

Findings

With high probability, hi_D(\u03b3)  hi() + 1

The result applies to Cayley graphs over certain abelian groups

Provides bounds on automorphism-respecting colorings in random Cayley graphs

Abstract

The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $\chi_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $\phi$ of the graph $G$ fixes each color class of $G$. In this paper, we consider random Cayley graphs $\Gamma(A,S)$ defined over certain abelian groups $A$ and show that with probability at least $1-n^{-\Omega(\log n)}$ we have, $\chi_D(\Gamma)\le\chi(\Gamma) + 1$.