An Oriented Version of the 1-2-3 Conjecture

Discussiones Mathematicae Graph Theory

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