Distinguishing infinite star-free graphs

TL;DR

This paper extends the understanding of distinguishing colourings in infinite star-free graphs, showing that certain locally finite graphs without induced stars can be distinguished with a limited number of colours.

Contribution

It generalizes previous finite results to all locally finite connected star-free graphs of order at least six.

Findings

Infinite locally finite star-free graphs admit distinguishing colourings with at most n-1 colours.

Extension of finite graph results to infinite graphs.

Provides conditions under which infinite graphs can be distinguished.

Abstract

Call a colouring of a graph \emph{distinguishing} if the only automorphism of this graph which preserves said colouring is the identity. Let $H$ be an arbitrary graph. We say that a graph $G$ is \emph{$H$-free} if $G$ does not contain an induced subgraph isomorphic to $H$. Kargul, Musia{\l}, Pal and Gorzkowska showed that if $n$ is a natural number greater than two, then every finite connected $K_{1,n}$-free graph of order at least six admits a distinguishing edge colouring with at most $n-1$ colours. We extend this result to all locally finite connected $K_{1,n}$-free graphs of order at least six.