An Injective Version of the 1-2-3 Conjecture

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 \(1,\dots ,\Delta (G)\). Towards this speculation, we provide bounds on the maximum label value needed in general. In particular, we prove that using labels \(1,\dots ,\Delta (G)\) is indeed sufficient when G is a tree, a particular cactus, or when its injective chromatic number \(\mathrm{\chi _{\mathrm{i}}}(G)\) is equal to \(\Delta (G)\).

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Funding

Julien Bensmail was supported by a funding granted by the program “Jeunes Talents FRANCE-CHINE”. Bi Li was supported by the National Natural Science Foundation of China (Nos. 11701440, 11626181). Binlong Li was supported by the National Natural Science Foundation of China (No. 11601429) and the Fundamental Research Funds for the Central Universities (No. 3102019ghjd003).

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Correspondence to Julien Bensmail.

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Bensmail, J., Li, B. & Li, B. An Injective Version of the 1-2-3 Conjecture. Graphs and Combinatorics (2020). https://doi.org/10.1007/s00373-020-02252-y

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Keywords

  • Proper labelling
  • Injective vertex-colouring
  • 1-2-3 Conjecture