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An Oriented Version of the 1-2-3 Conjecture

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Baudon O., Bensmail J., Sopena É.

Discussiones Mathematicae Graph Theory

2015

tom 35

nr 1

141-156

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.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Baudon O.

1 Bensmail J.

1 Sopena É.

1 2015

Wyniki wyszukiwania

An Oriented Version of the 1-2-3 Conjecture