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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 / ∈ {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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Path spectra for trees

Chen

Faudree

Šoltés³

2010

Discrete Mathematics

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Cited by 5 publications

References 59 publications

Decomposing highly connected graphs into paths of length five

Decomposing highly connected graphs into paths of length five

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

Path spectra for trees

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