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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
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
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).
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Submitted on : Wednesday, January 29, 2020 - 1:08:21 PM
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Julien Bensmail, Bi Li, Binlong Li. An injective version of the 1-2-3 Conjecture. [Research Report] Université côte d'azur. 2020. ⟨hal-02459377⟩

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