Discussiones Mathematicae Graph Theory 30(4) (2010) 591-609
doi: 10.7151/dmgt.1516

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ON THE EXISTENCE OF A CYCLE OF LENGTH AT LEAST 7 IN A (1, 2)-TWIN-FREE GRAPH

David Auger,  Irène Charon,  Olivier Hudry

Institut Telecom - Telecom ParisTech & Centre National
de la Recherche Scientifique - LTCI UMR 5141
46, rue Barrault, 75634 Paris Cedex 13, France

Antoine Lobstein

Centre National de la Recherche Scientifique - LTCI UMR 5141
& Telecom ParisTech
46, rue Barrault, 75634 Paris Cedex 13, France
e-mail: {david.auger, irene.charon, olivier.hudry,
  antoine.lobstein}@telecom-paristech.fr

Abstract

We consider a simple, undirected graph G. The ball of a subset Y of vertices in G is the set of vertices in G at distance at most one from a vertex in Y. Assuming that the balls of all subsets of at most two vertices in G are distinct, we prove that G admits a cycle with length at least 7.

Keywords: undirected graph, twin subsets, identifiable graph, distinguishable graph, identifying code, maximum length cycle.

2010 Mathematics Subject Classification: 05C38, 05C75.

References

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Received 27 July 2009
Revised 14 December 2009
Accepted 14 December 2009