Skip to Main content Skip to Navigation
Journal articles

An injective version of the 1-2-3 Conjecture

Julien Bensmail 1 Bi Li 2 Binlong Li 3
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : In this work, we introduce and study a new graph labelling problem standing as a combination of the 1-2-3 Conjecture and injective colouring of graphs, which also finds connections with the notion of graph irregularity. In this problem, the goal, given a graph G, is to label the edges of G so that every two vertices having a common neighbour get incident to different sums of labels. We are interested in the minimum k such that G admits such a k-labelling. We suspect that almost all graphs G can be labelled this way using labels 1,...,∆(G). Towards this speculation, we provide bounds on the maximum label value needed in general. In particular, we prove that using labels 1,...,∆(G) is indeed sufficient when G is a tree, a particular cactus, or when its injective chromatic number χi(G) is equal to ∆(G).
Document type :
Journal articles
Complete list of metadata

Cited literature [16 references]  Display  Hide  Download
Contributor : Julien Bensmail <>
Submitted on : Wednesday, January 29, 2020 - 1:08:21 PM
Last modification on : Sunday, January 17, 2021 - 1:57:34 AM
Long-term archiving on: : Thursday, April 30, 2020 - 3:50:45 PM


Files produced by the author(s)


  • HAL Id : hal-02459377, version 1



Julien Bensmail, Bi Li, Binlong Li. An injective version of the 1-2-3 Conjecture. Graphs and Combinatorics, Springer Verlag, 2021, 37, pp.281-311. ⟨hal-02459377⟩



Record views


Files downloads