In the random k-uniform hypergraph Hn,p;k of order n each possible k-tuple appears independently with probability p. A loose Hamilton cycle is a cycle of order n in which every pair of adjacent edges intersects in a single vertex. We prove that if pnk−1/ log n tends to infinity with n then lim n→∞ 2(k−1)|n Pr(Hn,p;k contains a loose Hamilton cycle) = 1. This is asymptotically best possible.

Let n ≥ 3 and λ ≥ 1 be integers. Let λKn denote the complete multigraph with edge-multiplicity λ. In this paper, we show that there exists a symmetric Hamilton cycle decomposition of λK2m for all even λ ≥ 2 and m ≥ 2. Also we show that there exists a symmetric Hamilton cycle decomposition of λK2m − F for all odd λ ≥ 3 and m ≥ 2. In fact, our results together with the earlier results (by Walecki...

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graphs and hypergraphs. In particular, we firstly show that for k > 3, if pnk−1/ log n tends to infinity, then a random k-uniform hypergraph on n vertices, with edge probability p, with high probability (w.h.p.) contains a l...

In the random k-uniform hypergraph H (k) n,p of order n, each possible k-tuple appears independently with probability p. A loose Hamilton cycle is a cycle of order n in which every pair of consecutive edges intersects in a single vertex. It was shown by Frieze that if p ≥ c(log n)/n2 for some absolute constant c > 0, then a.a.s. H (3) n,p contains a loose Hamilton cycle, provided that n is divi...

The kth power of a cycle is obtained by adding an edge between all pairs of vertices whose distance on the cycle is at most k. In 1962, Pósa conjectured that a graph G on n vertices contains a square of a Hamilton cycle if it has minimum degree δ(G) ≥ 3 n, and, 11 years later, Seymour claimed that δ(G) ≥ k k+1 n is sufficient for the appearance of a kth power of a Hamilton cycle for any k ≥ 2. ...

in this paper, we prove that the wrapped Butterfly graph WBF(d; n) of degree d and dimension n is decomposable into Hamilton cycles. This answers a conjecture of D. Barth and A. Raspaud who solved the case d = 2. Key-words: Butterfly graph, graph theory, Hamiltonism, Hamilton decomposition, Hamilton cycle, Hamilton circuit, perfect matching. (Résumé : tsvp) This work has been supported by the C...