Local finiteness, distinguishing numbers and Tucker's conjecture

TL;DR

This paper investigates Tucker's conjecture on distinguishing colourings in graphs, showing local finiteness is essential by providing a counterexample in non-locally finite graphs.

Contribution

The paper proves the necessity of local finiteness in Tucker's conjecture by constructing a non-locally finite graph that cannot be distinguished with finitely many colours.

Findings

Local finiteness is necessary for Tucker's conjecture.

Constructed a non-locally finite graph with no finite distinguishing colourings.

Supports the conjecture's conditions with a counterexample.

Abstract

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a non-locally finite graph for which no finite number of colours suffices.