Skip to Main content Skip to Navigation
Journal articles

1-2-3 Conjecture in Digraphs: More Results and Directions

Julien Bensmail 1 Kasper Lyngsie 2
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Horňak, Przybyło and Woźniak recently proved that, a small class of obvious exceptions apart, every digraph can be 4-arc-weighted so that, for every arc u->v, the sum of weights incoming to u is different from the sum of weights outgoing from v. They conjectured a stronger result, namely that the same statement with 3 instead of 4 should also be true. We verify this conjecture in this work. This work takes place in a recent "quest" towards a directed version of the 1-2-3 Conjecture, the variant above being one of the last introduced ones. We take the occasion of this work to establish a summary of all results known in this field, covering known upper bounds, complexity aspects, and choosability. On the way we prove additional results which were missing in the whole picture. We also mention the aspects that remain open.
Complete list of metadatas

Cited literature [17 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02269482
Contributor : Julien Bensmail <>
Submitted on : Monday, March 2, 2020 - 1:54:17 PM
Last modification on : Thursday, March 5, 2020 - 12:20:46 PM

File

inverse-luczak.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02269482, version 2

Collections

Citation

Julien Bensmail, Kasper Lyngsie. 1-2-3 Conjecture in Digraphs: More Results and Directions. Discrete Applied Mathematics, Elsevier, In press. ⟨hal-02269482v2⟩

Share

Metrics

Record views

30

Files downloads

31