On the distinguishing chromatic number in hereditary graph classes

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 ${\chi }_D(G)\le 2\mathrm{\Delta }(G)$ 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.