# An Oriented Version of the 1-2-3 Conjecture

• Volume: 35, Issue: 1, page 141-156
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Olivier Baudon, Julien Bensmail, and Éric Sopena. "An Oriented Version of the 1-2-3 Conjecture." Discussiones Mathematicae Graph Theory 35.1 (2015): 141-156. <http://eudml.org/doc/271217>.

@article{OlivierBaudon2015,
abstract = {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.},
author = {Olivier Baudon, Julien Bensmail, Éric Sopena},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {oriented graph; neighbour-sum-distinguishing arc-weighting; complexity; 1-2-3 Conjecture; 1-2-3 conjecture},
language = {eng},
number = {1},
pages = {141-156},
title = {An Oriented Version of the 1-2-3 Conjecture},
url = {http://eudml.org/doc/271217},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Olivier Baudon
AU - Julien Bensmail
AU - Éric Sopena
TI - An Oriented Version of the 1-2-3 Conjecture
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 1
SP - 141
EP - 156
AB - 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.
LA - eng
KW - oriented graph; neighbour-sum-distinguishing arc-weighting; complexity; 1-2-3 Conjecture; 1-2-3 conjecture
UR - http://eudml.org/doc/271217
ER -

## References

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1. [1] B. Seamone, The 1-2-3 Conjecture and related problems: a survey, Technical Report, available at http://arxiv.org/abs/1211.5122 (2012).
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9. [9] M. Khatirinejad, R. Naserasr, M. Newman, B. Seamone and B. Stevens, Digraphs are 2-weight choosable, Electron. J. Combin. 18 (2011) #1. Zbl1205.05101
10. [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] Zbl1042.05045
11. [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. Zbl1315.05105
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