Distinguishing symmetric digraphs by proper arc-colourings of type I

TL;DR

This paper establishes an optimal upper bound on the number of colours needed for proper arc-colourings of symmetric digraphs to be distinguishing, advancing understanding of graph automorphisms and colourings.

Contribution

It introduces and proves the optimal upper bound for the minimum colours required in a distinguishing proper arc-colouring of type I for symmetric digraphs.

Findings

Optimal upper bound of ⌈2√Δ(G)⌉ colours for distinguishing proper arc-colourings.

The same bound is optimal for a related colouring type with monochromatic 2-path restrictions.

Provides new insights into automorphism-distinguishing colourings of symmetric digraphs.

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.