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 $\Delta(G)\leq d$ 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 $K_{1,3}$, 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.
Highlights
We consider countable graphs, that is, graphs with finite or denumerable vertex sets, and use standard graph theoretic notation
We prove the following result: if G is a connected, countable graph of maximum degree at most 4, without an induced claw K1,3, D(G) = 2 whenever m(G) > 2, with three exceptional small graphs
The asymmetric colouring number or distinguishing number D(G) of a graph G is the least number of colours in an asymmetric vertex colouring, and the distinguishing
Summary
That is, graphs with finite or denumerable vertex sets, and use standard graph theoretic notation. A natural extension of the result of Lehner would be a confirmation of the Infinite Motion Conjecture for claw-free graphs Another motivation for our research was the following question posed by Babai in 2018. This question for d = 3 was fully answered in a recent paper of Huning, Imrich, Kloas, Schreiber and Tucker [7], who gave a complete classification of all countable connected graphs G of maximum degree ∆(G) = 3 and distinguishing number D(G) 3 As a consequence, they observed that the only subcubic graphs G with D(G) > 2 and m(G) > 2 are the cube K22K22K2 and the Petersen graph, both containing an induced claw.
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