An Oriented Version of the 1-2-3 Conjecture

• Autorzy: Olivier Baudon , Julien Bensmail , Éric Sopena
• Języki publikacji: EN

Abstrakty

EN

The well-known 1-2-3 Conjecture addressed by Karoński, Luczak and Thomason asks whether the edges of every undirected graph G with no isolated edge can be assigned weights from {1, 2, 3} so that the sum of incident weights at each vertex yields a proper vertex-colouring of G. In this work, we consider a similar problem for oriented graphs. We show that the arcs of every oriented graph −G⃗ can be assigned weights from {1, 2, 3} so that every two adjacent vertices of −G⃗ receive distinct sums of outgoing weights. This result is tight in the sense that some oriented graphs do not admit such an assignment using the weights from {1, 2} only. We finally prove that deciding whether two weights are sufficient for a given oriented graph is an NP-complete problem. These results also hold for product or list versions of this problem.

Słowa kluczowe

EN

oriented graph; neighbour-sum-distinguishing arc-weighting; complexity; 1-2-3 Conjecture

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2015
• Tom: 35
• Numer: 1
• Strony: 141-156

Daty

wydano

2015-02-01

otrzymano

2013-10-08

poprawiono

2014-03-17

zaakceptowano

2014-04-29

online

2015-02-06

Twórcy

autor

Olivier Baudon

• Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France CNRS, LaBRI, UMR 5800, F-33400 Talence, France

autor

Julien Bensmail

• Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France CNRS, LaBRI, UMR 5800, F-33400 Talence, France

autor

Éric Sopena

• Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France CNRS, LaBRI, UMR 5800, F-33400 Talence, France

Bibliografia

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• [9] M. Khatirinejad, R. Naserasr, M. Newman, B. Seamone and B. Stevens, Digraphs are 2-weight choosable, Electron. J. Combin. 18 (2011) #1.
• [10] M. Karoński, T. Luczak and A. Thomason, Edge weights and vertex colours, J. Combin. Theory (B) 91 (2004) 151-157. doi:10.1016/j.jctb.2003.12.001[Crossref]
• [11] O. Baudon, J. Bensmail, J. Przyby lo and M. Wózniak, On decomposing regular graphs into locally irregular subgraphs, Preprint MD 065 (2012), available at http://www.ii.uj.edu.pl/preMD/index.php.
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Identyfikatory

DOI

10.7151/dmgt.1791