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

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

 [1] D. Auger, Induced paths in twin-free graphs, Electron. J. Combinatorics 15 (2008) N17. [2] C. Berge, Graphes (Gauthier-Villars, 1983). [3] C. Berge, Graphs (North-Holland, 1985). [4] I. Charon, I. Honkala, O. Hudry and A. Lobstein, Structural properties of twin-free graphs, Electron. J. Combinatorics 14 (2007) R16. [5] I. Charon, O. Hudry and A. Lobstein, On the structure of identifiable graphs: results, conjectures, and open problems, in: Proceedings 29th Australasian Conference in Combinatorial Mathematics and Combinatorial Computing (Taupo, New Zealand, 2004) 37-38. [6] R. Diestel, Graph Theory (Springer, 3rd edition, 2005). [7] S. Gravier and J. Moncel, Construction of codes identifying sets of vertices, Electron. J. Combinatorics 12 (2005) R13. [8] I. Honkala, T. Laihonen and S. Ranto, On codes identifying sets of vertices in Hamming spaces, Designs, Codes and Cryptography 24 (2001) 193-204, doi: 10.1023/A:1011256721935. [9] T. Laihonen, On cages admitting identifying codes, European J. Combinatorics 29 (2008) 737-741, doi: 10.1016/j.ejc.2007.02.016. [10] T. Laihonen and J. Moncel, On graphs admitting codes identifying sets of vertices, Australasian J. Combinatorics 41 (2008) 81-91. [11] T. Laihonen and S. Ranto, Codes identifying sets of vertices, in: Lecture Notes in Computer Science, No. 2227 (Springer-Verlag, 2001) 82-91. [12] A. Lobstein, Bibliography on identifying, locating-dominating and discriminating codes in graphs, http://www.infres.enst.fr/ ~ lobstein/debutBIBidetlocdom.pdf. [13] J. Moncel, Codes identifiants dans les graphes, Thèse de Doctorat, Université de Grenoble, France, 165 pages, June 2005.