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Disjoint cycles of different lengths in graphs and digraphs. (English) Zbl 1376.05038
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.

05C12 Distance in graphs
05C07 Vertex degrees
05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
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