Breaking graph symmetries by edge colourings

Lehner, Florian

doi:10.1016/j.jctb.2017.06.001

Cited by 20 publications

(15 citation statements)

References 16 publications

(20 reference statements)

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“…Partial results were obtained in . Let us mentioned that the Infinite Edge‐Motion Conjecture by Broere and Pilśniak was proved by Lehner in , thus confirming the Infinite Motion Conjecture for line graphs.…”

Section: Introductionmentioning

confidence: 65%

The optimal general upper bound for the distinguishing index of infinite graphs

Pilśniak

Stawiski

2019

Journal of Graph Theory

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The distinguishing index D′(G) of a graph G is the least cardinal number d such that G has an edge‐coloring with d colors, which is preserved only by the trivial automorphism. We prove a general upper bound D′(G)goodbreakinfix≤normalΔgoodbreakinfix−1 for any connected infinite graph G with finite maximum degree normalΔgoodbreakinfix≥3. This is in contrast with finite graphs since for every normalΔgoodbreakinfix≥3 there exist infinitely many connected, finite graphs G with D′(G)goodbreakinfix=normalΔ. We also give examples showing that this bound is sharp for any maximum degree normalΔ.

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Section: Introductionmentioning

confidence: 65%

The optimal general upper bound for the distinguishing index of infinite graphs

Pilśniak

Stawiski

2019

Journal of Graph Theory

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“…They proved that for every connected infinite graph we have D (G) ∆, where ∆ is a cardinal number such that degree of any vertex is at most ∆. They also stated an infinite edge-motion conjecture (analogous to the Infinite Motion Conjecture of Tucker) that was very recently confirmed by Lehner [10]. Kalinowski and Pilśniak [7] also compared D(G) and D (G) when G is a finite graph.…”

Section: Introductionmentioning

confidence: 96%

Bounds for Distinguishing Invariants of Infinite Graphs

Imrich

Kalinowski

Pilśniak

et al. 2017

Electron. J. Combin.

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We consider infinite graphs. The distinguishing number D(G) of a graph G is the minimum number of colours in a vertex colouring of G that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by D (G). We prove that D (G) D(G) + 1. For proper colourings, we study relevant invariants called the distinguishing chromatic number χ D (G), and the distinguishing chromatic index χ D (G), for vertex and edge colourings, respectively. We show that χ D (G) 2∆(G) − 1 for graphs with a finite maximum degree ∆(G), and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that χ D (G) χ (G) + 1, where χ (G) is the chromatic index of G, and we prove a similar result χ D (G) χ (G) + 1 for proper total colourings. A number of conjectures are formulated.

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“…Generally D (G) can be arbitrary smaller than D(G), for instance if p ≥ 6, then D (K p ) = 2 and D(K p ) = p. Conversely, there is an upper bound on D (G) in terms of D(G). In [12,Theorem 11] (see also [17,Theorem 8] for an alternative proof) it is proved that if G is a connected graph of order at least 3, then…”

Section: Introductionmentioning

confidence: 99%