an:06817440
Zbl 1376.05038
Bensmail, Julien; Harutyunyan, Ararat; Le, Ngoc Khang; Li, Binlong; Lichiardopol, Nicolas
Disjoint cycles of different lengths in graphs and digraphs
EN
Electron. J. Comb. 24, No. 4, Research Paper P4.37, 24 p. (2017).
00370856
2017
j
05C12 05C07 05C20 05C38
vertex-disjoint cycles; different lengths; minimum degree
Summary: 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 \(k \geqslant 1\), 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.