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arXiv · 2505.17193

On the distinguishing chromatic number in hereditary graph classes

Christoph Brause·Rafał Kalinowski·Monika Pilśniak·Ingo Schiemeyer
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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.

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Christoph Brause, Rafał Kalinowski, Monika Pilśniak, Ingo Schiemeyer. 2025-05-22. On the distinguishing chromatic number in hereditary graph classes. https://arxiv.org/abs/2505.17193

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