Disjoint cycles of different lengths in graphs and digraphs

Abstract : In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has~$k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider other cases such as when the graph is triangle-free, or the $k$ cycles are required to have different lengths modulo some value $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for some regular digraphs and digraphs of small order.
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Electronic Journal of Combinatorics, Electronic Journal of Combinatorics, 2017
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Dernière modification le : lundi 4 décembre 2017 - 15:14:19

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  • HAL Id : hal-01653334, version 1

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Julien Bensmail, Ararat Harutyunyan, Ngoc Khang Le, Binlong Li, Nicolas Lichiardopol. Disjoint cycles of different lengths in graphs and digraphs. Electronic Journal of Combinatorics, Electronic Journal of Combinatorics, 2017. 〈hal-01653334〉

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