TY - JOUR

T1 - On asymmetric colourings of claw-free graph

AU - Imrich, Wilfried

AU - Kalinowski, Rafal

AU - Pilśniak, Monika

AU - Woźniak, Mariusz

N1 - Publisher Copyright: © The authors. Released under the CC BY-ND license (International 4.0).

PY - 2021/7/16

Y1 - 2021/7/16

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85110712632&partnerID=8YFLogxK

U2 - 10.37236/8886

DO - 10.37236/8886

M3 - Article

AN - SCOPUS:85110712632

VL - 28.2021

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 3

M1 - P3.25

ER -