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Journal Articles The Electronic Journal of Combinatorics Year : 2017

## Disjoint cycles of different lengths in graphs and digraphs

Ngoc Khang Le
• Function : Author
Binlong Li
• Function : Author
Nicolas Lichiardopol
• Function : Author

#### 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.

#### Domains

Computer Science [cs] Discrete Mathematics [cs.DM]

### Dates and versions

hal-01653334 , version 1 (01-12-2017)

### Identifiers

• HAL Id : hal-01653334 , version 1

### Cite

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, 2017, 24 (4). ⟨hal-01653334⟩

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