Nicolas criterion for the Riemann Hypothesis is based on an inequality that Euler totient function must satisfy at primorial numbers. A natural approach to derive this inequality would be to prove that a specific sequence related to that bound is strictly decreasing. We show that, unfortunately, this latter fact would contradict Cramér conjecture on gaps between consecutive primes. An analogous situation holds when replacing Euler totient by Dedekind Ψ function.

In 1981, Bermond and Thomassen conjectured that every digraph with minimum out-degree at least 2k − 1 contains k disjoint cycles. This conjecture is trivial for k = 1, and was established for k = 2 by Thomassen in 1983. We verify it for the next case, by proving that every digraph with minimum out-degree at least five contains three disjoint cycles. To show this, we improve Thomassen's result by proving that every digraph whose vertices have out-degree at least three, except at most two with out-degree two, indeed contains two disjoint cycles.

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r − 1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r = 2, and very recently the conjecture was proved for the case where r = 3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we present two proofs of this conjecture for tournaments with minimum in-degree at least 2r − 1. In particular, this shows that the conjecture is true for almost regular tournament. In the first proof, we prove auxiliary results about union of sets contained in other union of sets, that might be of independent interest. The second one uses a more graph-theoretical approach, by studying the properties of a maximum set of vertex-disjoint directed triangles.

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. 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 ≥ 1, every graph with minimum degree at least k 2 +5k−2 2 has k vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider stronger situations, and exhibit degree bounds (some of which are best possible) when e.g. the graph is triangle-free, or the k cycles are requested to have different lengths congruent to some values modulo some 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 regular digraphs and digraphs of small order. 2 for every k (Theorem 2.12). Several more constrained situations are then considered, e.g. when the graph is trianglefree or the vertex-disjoint cycles are requested to be more than just of different lengths; in these situations as well, we exhibit bounds (most of which are tight) on the least minimum degree required to guarantee the existence of the k desired vertex-disjoint cycles. We also consider the opposite direction, and conjecture that for every D ≥ 3, every graph G verifying k + 1 ≤ δ(G) ≤ ∆(G) ≤ D and of large enough order has k vertex-disjoint cycles of different lengths (see Conjecture 2.18). To support this conjecture, we prove it for k = 2 (Theorem 2.19). This in particular yields that every cubic graph of order more than 14 has two vertex-disjoint cycles of different lengths, which is tight (see Theorem 2.20).We then consider, in Section 3, the same question for digraphs: What minimum outdegree is required for a digraph to have at least k vertex-disjoint directed cycles of different lengths? The existence of such a minimum out-degree was conjectured by Lichiardopol in [9], who verified it for k = 2. We here give further support to Lichiardopol's Conjecture by showing it to hold for tournaments (see Corollary 3.6), and, using the probabilistic method, for regular digraphs (Theorem 3.10) and digraphs of small order (Theorem 3.11). Disjoint cycles of different lengths in undirected graphsIn this section, we consider the existence of disjoint cycles of different lengths in graphs. We start off, in Section 2.1, by recalling a few results and introducing new results and concepts of independent interest. We then prove our main results in Section 2.2. PreliminariesLet G be a graph and X a subset of V (G). We use G[X] to denote the subgraph of G induced by X, and G − X to denote the subgraph of G induced by V (G)\X. For two disjoint subsets X,

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