Distinguishing symmetric digraphs by proper arc-colourings of type I
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arXiv
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| Format: | Preprint |
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2025
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| author | Kalinowski, Rafał Pilśniak, Monika Prorok, Magdalena |
| author_facet | Kalinowski, Rafał Pilśniak, Monika Prorok, Magdalena |
| contents | 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 $\lceil 2\sqrt{Δ(G)}\rceil$ 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 $\lceil 2\sqrt{Δ(G)}\rceil$ is optimal for another type of proper colouring of $\overleftrightarrow{G}$, when only monochromatic 2-paths are forbidden. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13979 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinguishing symmetric digraphs by proper arc-colourings of type I Kalinowski, Rafał Pilśniak, Monika Prorok, Magdalena Combinatorics 05C15, 05C20, 05C25 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 $\lceil 2\sqrt{Δ(G)}\rceil$ 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 $\lceil 2\sqrt{Δ(G)}\rceil$ is optimal for another type of proper colouring of $\overleftrightarrow{G}$, when only monochromatic 2-paths are forbidden. |
| title | Distinguishing symmetric digraphs by proper arc-colourings of type I |
| topic | Combinatorics 05C15, 05C20, 05C25 |
| url | https://arxiv.org/abs/2506.13979 |