Decomposability of graphs into subgraphs fulfilling the 1-2-3 Conjecture

Julien Bensmail 1 Jakub Przybyƚo 2
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every d-regular graph, d ≥ 2, can be decomposed into at most 2 subgraphs (without isolated edges) fulfilling the 1-2-3 Conjecture if d not in {10, 11, 12, 13, 15, 17}, and into at most 3 such subgraphs in the remaining cases. Additionally, we prove that in general every graph without isolated edges can be decomposed into at most 24 subgraphs fulfilling the 1-2-3 Conjecture, improving the previously best upper bound of 40. Both results are partly based on applications of the Lovász Local Lemma.
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Julien Bensmail, Jakub Przybyƚo. Decomposability of graphs into subgraphs fulfilling the 1-2-3 Conjecture. Discrete Applied Mathematics, Elsevier, 2019, ⟨10.1016/j.dam.2019.04.011⟩. ⟨hal-02288797⟩

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