Discussiones Mathematicae Graph Theory | |
An Oriented Version of the 1-2-3 Conjecture | |
Bensmail Julien1  Sopena Éric1  Baudon Olivier1  | |
[1] Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France CNRS, LaBRI, UMR 5800, F-33400 Talence, France; | |
关键词: oriented graph; neighbour-sum-distinguishing arc-weighting; complexity; 1-2-3 conjecture; | |
DOI : 10.7151/dmgt.1791 | |
来源: DOAJ |
【 摘 要 】
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.
【 授权许可】
Unknown