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.
Highlights
In this paper, we study degree conditions guaranteeing the existence in a graph of a certain number of vertex-disjoint cycles whose lengths satisfy particular properties
We consider, given some k 1, the minimum degree required for a graph to have at least k vertex-disjoint cycles of different lengths
The central question of this work is the following: What is the smallest minimum degree f (k) required so that a graph with minimum degree f (k) has at least k vertex-disjoint cycles of different lengths? We answer this question in the following result
Summary
We study degree conditions guaranteeing the existence in a graph We consider, given some k 1, the minimum degree required for a graph to have at least k vertex-disjoint cycles of different lengths (and sometimes additional length properties). We show that this value is precisely k2+5k−2 2 for every k D 3, every graph G of large enough order verifying k + 1 δ(G) ∆(G) D has k vertex-disjoint cycles of different lengths (see Conjecture 20) To support this conjecture, we prove it for k = 2 (Theorem 23). We prove it for k = 2 (Theorem 23) This in particular yields that every cubic graph of order more than 14 has two vertex-disjoint cycles of different lengths, which is tight We here give further support to Lichiardopol’s Conjecture by showing it to hold for tournaments (see Corollary 30), and, using the probabilistic method, for some regular digraphs (Theorem 34) and digraphs of small order (Theorem 35)
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