# Disjoint cycles of different lengths in graphs and digraphs

1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
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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Article dans une revue
Electronic Journal of Combinatorics, Electronic Journal of Combinatorics, 2017, 24 (4)

Littérature citée [14 références]

https://hal.archives-ouvertes.fr/hal-01653334
Contributeur : Julien Bensmail <>
Soumis le : vendredi 1 décembre 2017 - 12:21:34
Dernière modification le : vendredi 20 avril 2018 - 15:44:26

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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, 24 (4). 〈hal-01653334〉

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