Edge Weights and Vertex Colours: Minimizing Sum Count - Archive ouverte HAL Access content directly
Journal Articles Discrete Applied Mathematics Year : 2019

Edge Weights and Vertex Colours: Minimizing Sum Count


Neighbour-sum-distinguishing edge-weightings are a way to "encode" proper vertex-colourings via the sums of weights incident to the vertices. Over the last decades, this notion has been attracting, in the context of several conjectures, ingrowing attention dedicated, notably, to understanding, which weights are needed to produce neighbour-sum-distinguishing edge-weightings for a given graph. This work is dedicated to investigating another related aspect, namely the minimum number of distinct sums/colours we can produce via a neighbour-sum-distinguishing edge-weighting of a given graph G, and the role of the assigned weights in that context. Clearly, this minimum number is bounded below by the chromatic number χ(G) of G. When using weights of ℤ, we show that, in general, we can produce neighbour-sum-distinguishing edge-weightings generating χ(G) distinct sums, except in the peculiar case where G is a balanced bipartite graph, in which case χ(G)+1 distinct sums can be generated. These results are best possible. When using k consecutive weights 1,...,k, we provide both lower and upper bounds, as a function of the maximum degree ∆, on the maximum least number of sums that can be generated for a graph with maximum degree ∆. For trees, which, in general, admit neighbour-sum-distinguishing 2-edge-weightings, we prove that this maximum, when using weights 1 and 2, is of order 2 log_2 ∆. Finally, we also establish the NP-hardness of several decision problems related to these questions.
Fichier principal
Vignette du fichier
123sums-07062019.pdf (415.31 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-01839537 , version 1 (15-07-2018)
hal-01839537 , version 2 (07-06-2019)


  • HAL Id : hal-01839537 , version 2


Olivier Baudon, Julien Bensmail, Hervé Hocquard, Mohammed Senhaji, Eric Sopena. Edge Weights and Vertex Colours: Minimizing Sum Count. Discrete Applied Mathematics, 2019, 270, pp.13-24. ⟨hal-01839537v2⟩
609 View
656 Download


Gmail Facebook X LinkedIn More