Year: 2025
Type: article
Abstract: A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is an asymmetric colouring with 2 colours. This conjecture was recently confirmed by Babai, using the heavy machinery of the classification of finite simple groups. We make progress towards a purely combinatorial proof of this conjecture in the special case of graphs with bounded maximal degree. More precisely, using only elementary combinatorial methods, we prove that if every automorphism of a connected graph with maximal degree Δ moves infinitely many vertices, then there is an asymmetric colouring using O(sqrt(Δ) log(Δ)) colours. This is the first improvement which does not depend on the classification of finite simple groups over the trivial bound of O(Δ).
Authors: Lehner, Florian, Pilśniak, Monika, Stawiski, Marcin
Institutions: Graz University of Technology, AGH University of Krakow
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funders: Austrian Science Fund, OeAD-GmbH
Awards: Graphs and groups, G8377390777
OpenAlex ID: W7115272404
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