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2015 | 35 | 1 | 141-156
Tytuł artykułu

An Oriented Version of the 1-2-3 Conjecture

Treść / Zawartość
Warianty tytułu
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.
Wydawca
Rocznik
Tom
35
Numer
1
Strony
141-156
Opis fizyczny
Daty
wydano
2015-02-01
otrzymano
2013-10-08
poprawiono
2014-03-17
zaakceptowano
2014-04-29
online
2015-02-06
Twórcy
  • Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France CNRS, LaBRI, UMR 5800, F-33400 Talence, France, olivier.baudon@labri.fr
  • Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France CNRS, LaBRI, UMR 5800, F-33400 Talence, France, julien.bensmail@labri.fr
autor
  • Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France CNRS, LaBRI, UMR 5800, F-33400 Talence, France, eric.sopena@labri.fr
Bibliografia
  • [1] B. Seamone, The 1-2-3 Conjecture and related problems: a survey, Technical Report, available at http://arxiv.org/abs/1211.5122 (2012).
  • [2] W. Imrich and S. Klavzar, Product Graphs: Structure and Recognition (Wiley- Interscience, New York, 2000).
  • [3] J.W. Moon, Topics on Tournaments (Holt, Rinehart and Winston, 1968).
  • [4] J. Skowronek-Kaziów, 1, 2 conjecture-the multiplicative version, Inform. Process. Lett. 107 (2008) 93-95. doi:10.1016/j.ipl.2008.01.006[Crossref]
  • [5] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W.H. Freeman, 1979).
  • [6] M. Kalkowski and M. Karoński and F. Pfender, Vertex-coloring edge-weightings: Towards the 1-2-3 conjecture, J. Combin. Theory (B) 100 (2010) 347-349. doi:10.1016/j.jctb.2009.06.002[WoS][Crossref]
  • [7] T. Bartnicki, J. Grytczuk and S. Niwczyk, Weight choosability of graphs, J. Graph Theory 60 (2009) 242-256. doi:10.1002/jgt.20354[Crossref]
  • [8] M. Borowiecki, J. Grytczuk and M. Pilśniak, Coloring chip configurations on graphs and digraphs, Inform. Process. Lett. 112 (2012) 1-4. doi:10.1016/j.ipl.2011.09.011[Crossref][WoS]
  • [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.
  • [12] L. Addario-Berry, R.E.L. Aldred, K. Dalal and B.A. Reed, Vertex colouring edge partitions, J. Combin. Theory (B) 94 (2005) 237-244. doi:10.1016/j.jctb.2005.01.001[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1791
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