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arXiv · 1803.07409

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

Julien Bensmail·Jakub Przybyło
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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, $d\geq 2$, can be decomposed into at most $2$ subgraphs (without isolated edges) fulfilling the 1-2-3 Conjecture if $d\notin\{10,11,12,13,15,17\}$, 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 Lov\'asz Local Lemma.

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Julien Bensmail, Jakub Przybyło. 2018-03-20. Decomposability of graphs into subgraphs fulfilling the 1-2-3 Conjecture. https://arxiv.org/abs/1803.07409

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