# Disjoint cycles of different lengths in graphs and digraphs

1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
3 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
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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Journal articles

Cited literature [14 references]

https://hal.archives-ouvertes.fr/hal-01653334
Contributor : Julien Bensmail <>
Submitted on : Friday, December 1, 2017 - 12:21:34 PM
Last modification on : Friday, June 25, 2021 - 3:40:06 PM

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distinct-lengths.pdf
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• HAL Id : hal-01653334, version 1

### Citation

Julien Bensmail, Ararat Harutyunyan, Ngoc Khang Le, Binlong Li, Nicolas Lichiardopol. Disjoint cycles of different lengths in graphs and digraphs. The Electronic Journal of Combinatorics, Open Journal Systems, 2017, 24 (4). ⟨hal-01653334⟩

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