Distinguishing Graphs of Maximum Valence 3

Hüning, Svenja; Imrich, Wilfried; Kloas, Judith; Schreber, Hannah; Tucker, Thomas W.

doi:10.37236/7281

Cited by 9 publications

(17 citation statements)

References 22 publications

(39 reference statements)

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“…The conjecture is open, though it was proved to be correct for several classes of graphs. For instance, Lehner, Pilśniak and Stawiski [14] recently confirmed this conjecture for graphs G of maximum degree ∆(G) 5 (the case ∆(G) = 3 was independently proved in [7]).…”

Section: Introductionmentioning

confidence: 85%

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On Asymmetric Colourings of Claw-Free Graphs

Imrich

Kalinowski

Pilśniak

et al. 2021

Electron. J. Combin.

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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.

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Section: Introductionmentioning

confidence: 85%

“…This question for d = 3 was fully answered in a recent paper of Hüning, 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)…”

Section: Introductionmentioning

confidence: 99%

On Asymmetric Colourings of Claw-Free Graphs

Imrich

Kalinowski

Pilśniak

et al. 2021

Electron. J. Combin.

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“…For n ≥ 3, the wreath graph W n is the lexicographic product C n [2K 1 ] of a cycle of length n with an edgeless graph of order 2, see Figure 1. It is easy to see that wreath graphs form an infinite family of connected exceptional 4-valent vertex-transitive graphs, thus providing a negative answer to [7,Question 2]. Our main result shows that this is the only such family, that is, apart from the wreath graphs, there are only finitely many connected exceptional 4-valent vertex-transitive graphs.…”

Section: Introductionmentioning

confidence: 91%

Distinguishing numbers of finite 4-valent vertex-transitive graphs

Lehner

Verret

2020

Ars Math. Contemp.

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“…We have answered Babai's question for trees using f (d) = 2 log 2 d . It is interesting to note that for maximum valence d = 3, the same function gives an asymmetric coloring with the exception of the cube and Petersen graph [5,Corollary 3.7].…”

Section: General Connected Graphsmentioning

confidence: 99%

Asymmetrizing trees of maximum valence $$2^{\aleph _0}$$

Imrich

Tucker

2020

Monatsh Math

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Let T be a finite or infinite tree and m the minimum number of vertices moved by the non-identity automorphisms of T. We give bounds on the maximum valence d of T that assure the existence of a vertex coloring of T with two colors that is preserved only by the identity automorphism. For finite m we obtain the bound $$d\le 2^{m/2}$$d≤2m/2 when T is finite, and $$d\le 2^{(m-2)/2}+2$$d≤2(m-2)/2+2 when T is infinite. For countably infinite m the bound is $$d\le 2^m.$$d≤2m. This relates to a question of Babai, who asked whether there existed a function f(d) such that every connected, locally finite graph G with maximum valence d has a 2-coloring of its vertices that is only preserved by the identity automorphism if the minimum number m of vertices moved by each non-identity automorphisms of G is at least $$m\ge f(d)$$m≥f(d). Our results give a positive answer for trees. The trees need not be locally finite, their maximal valence can be $$2^{\aleph _0}$$2ℵ0. For finite m we also extend our results to asymmetrizing trees by more than two colors.

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Distinguishing Graphs of Maximum Valence 3

Cited by 9 publications

References 22 publications

On Asymmetric Colourings of Claw-Free Graphs

On Asymmetric Colourings of Claw-Free Graphs

Distinguishing numbers of finite 4-valent vertex-transitive graphs

Asymmetrizing trees of maximum valence $$2^{\aleph _0}$$

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