On the distinguishing chromatic number in hereditary graph classes

TL;DR

This paper investigates how the maximum number of colours needed for a unique vertex colouring in graphs can be reduced within hereditary classes that exclude certain small induced subgraphs.

Contribution

It establishes that forbidding small induced subgraphs can significantly lower the upper bounds on the distinguishing chromatic number.

Findings

Upper bounds are reduced in hereditary graph classes.

The bound for $ ext{distinguishing chromatic number}$ depends on forbidden subgraphs.

Results extend understanding of symmetry-breaking colourings in graphs.

Abstract

The distinguishing chromatic number of a graph $G$, denoted $\chi_D(G)$, is the minimum number of colours in a proper vertex colouring of $G$ that is preserved by the identity automorphism only. Collins and Trenk proved that $\chi_D(G)\le 2\Delta(G)$ for any connected graph $G$, and the equality holds for complete balanced bipartite graphs $K_{p,p}$ and for $C_6$. In this paper, we show that the upper bound on $\chi_D(G)$ can be substantially reduced if we forbid some small graphs as induced subgraphs of $G$, that is, we study the distinguishing chromatic number in some hereditary graph classes.