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arXiv · 2506.13979

Distinguishing symmetric digraphs by proper arc-colourings of type I

Rafał Kalinowski·Monika Pilśniak·Magdalena Prorok
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Abstract

A symmetric digraph $\overleftrightarrow{G}$ is obtained from a simple graph $G$ by replacing each edge $uv$ with a pair of opposite arcs $\vec{uv}$, $\overrightarrow{vu}$. An arc-colouring $c$ of a digraph $\overleftrightarrow{G}$ is distinguishing if the only automorphism of $\overleftrightarrow{G}$ preserving the colouring $c$ is the identity. Behzad introduced the proper arc-colouring of type I as an arc-colouring such that any two consecutive arcs $\overrightarrow{uv}$, $\overrightarrow{vw}$ have distinct colours. We establish an optimal upper bound $\lceil 2\sqrt{\Delta(G)}\rceil$ for the least number of colours in a distinguishing proper colouring of type I of a connected symmetric digraph $\overleftrightarrow{G}$. Furthermore, we prove that the same upper bound $\lceil 2\sqrt{\Delta(G)}\rceil$ is optimal for another type of proper colouring of $\overleftrightarrow{G}$, when only monochromatic 2-paths are forbidden.

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Rafał Kalinowski, Monika Pilśniak, Magdalena Prorok. 2025-06-16. Distinguishing symmetric digraphs by proper arc-colourings of type I. https://arxiv.org/abs/2506.13979

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