On asymmetric colourings of claw-free graph
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- AGH University of Science and Technology Krakow
Abstract
A vertex colouring of a graph is asymmetric if it is preserved only by the identity automorphism. The minimum number of colours needed for an asymmetric colouring of a graph G is called the asymmetric colouring number or distinguishing number D(G) of G. It is well known that D(G) is closely related to the least number of vertices moved by any non-identity automorphism, the so-called motion m(G) of G. Large motion is usually correlated with small D(G). Recently, Babai posed the question whether there exists a function f (d) such that every connected, countable graph G with maximum degree ∆(G) d and motion m(G) > f (d) has an asymmetric 2-colouring, with at most finitely many exceptions for every degree. We prove the following result: if G is a connected, countable graph of maximum degree at most 4, without an induced claw K1,3, then D(G) = 2 whenever m(G) > 2, with three exceptional small graphs. This answers the question of Babai for d = 4 in the class of claw-free graphs.
Details
Original language | English |
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Article number | P3.25 |
Number of pages | 14 |
Journal | Electronic Journal of Combinatorics |
Volume | 28.2021 |
Issue number | 3 |
DOIs | |
Publication status | Published - 16 Jul 2021 |