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Article Dans Une Revue Discrete Applied Mathematics Année : 2019

A 1-2-3-4 result for the 1-2-3 Conjecture in 5-regular graphs

Résumé

The 1-2-3 Conjecture, posed by Karoński, Łuczak and Thomason, asks whether every connected graph G different from K2 can be 3-edge-weighted so that every two adjacent vertices of G get distinct sums of incident weights. Towards that conjecture, the best-known result to date is due to Kalkowski, Karoński and Pfender, who proved that it holds when relaxed to 5-edge-weightings. Their proof builds upon a weighting algorithm designed by Kalkowski for a total version of the problem. In this work, we present new mechanisms for using Kalkowski's algorithm in the context of the 1-2-3 Conjecture. As a main result we prove that every 5-regular graph admits a 4-edge-weighting that permits to distinguish its adjacent vertices via their incident sums.
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Dates et versions

hal-01509365 , version 1 (17-04-2017)
hal-01509365 , version 2 (01-10-2018)

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Julien Bensmail. A 1-2-3-4 result for the 1-2-3 Conjecture in 5-regular graphs. Discrete Applied Mathematics, 2019, 257, pp.31-39. ⟨10.1016/j.dam.2018.10.008⟩. ⟨hal-01509365v2⟩
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