On symmetries of edge and vertex colourings of graphs

TL;DR

This paper investigates the relationship between symmetries of edge and vertex colourings in graphs, showing how to find less symmetric colourings with the same number of colours and characterizing graphs based on their distinguishing indices.

Contribution

It introduces new results relating symmetries of edge and vertex colourings and characterizes graphs with specific distinguishing index properties.

Findings

For non-bicentred trees, every vertex colouring has a less symmetric edge colouring with the same number of colours.

For trees, every edge colouring has a less symmetric vertex colouring with the same number of edges.

Results help characterize graphs where the distinguishing index exceeds the distinguishing number.

Abstract

Let $c$ and $c'$ be edge or vertex colourings of a graph $G$. We say that $c'$ is less symmetric than $c$ if the stabiliser (in $\operatorname{Aut} G$) of $c'$ is contained in the stabiliser of $c$. We show that if $G$ is not a bicentred tree, then for every vertex colouring of $G$ there is a less symmetric edge colouring with the same number of colours. On the other hand, if $T$ is a tree, then for every edge colouring there is a less symmetric vertex colouring with the same number of edges. Our results can be used to characterise those graphs whose distinguishing index is larger than their distinguishing number.