Disjoint cycles of different lengths in graphs and digraphs - Archive ouverte HAL Access content directly
Journal Articles The Electronic Journal of Combinatorics Year : 2017

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.
Fichier principal
Vignette du fichier
distinct-lengths.pdf (362.86 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

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⟩
302 View
573 Download

Share

Gmail Facebook Twitter LinkedIn More