The famous Pósa conjecture states that every graph of minimum degree at least 2n/3 contains the square of a Hamilton cycle. This has been proved for large n by Komlós, Sarközy and Szemerédi. Here we prove that if p ≥ n−1/2+ε, then asymptotically almost surely, the binomial random graph Gn,p contains the square of a Hamilton cycle. This provides an ‘approximate threshold’ for the property in the...

We show that, for a natural notion of quasirandomness in k-uniform hypergraphs, any quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum vertex degree Ω(nk−1) contains a loose Hamilton cycle. We also give a construction to show that a k-uniform hypergraph satisfying these conditions need not contain a Hamilton `-cycle if k − ` divides k. The remaining values of ...

Paul Seymour conjectured that any graph G of order n and minimum degree at least k k+1n contains the k th power of a Hamilton cycle. We prove the following approximate version. For any > 0 and positive integer k, there is an n0 such that, if G has order n ≥ n0 and minimum degree at least ( k k+1 + )n, then G contains the kth power of a Hamilton cycle. c © 1998 John Wiley & Sons, Inc. J Graph Th...

Let m ≥ 2 and k ≥ 2 be integers. We show that K k×m has a decomposition into Hamilton cycles of Kierstead-Katona type if k | m. We also show that K (3) 3×m − T has a decomposition into Hamilton cycles where T is a 1-factor if and only if 3 m and m = 4. We introduce a notion of symmetry and comment on the existence of symmetric Hamilton cycle decompositions of K (k) k×m.

Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Stéphane Perennes* Thème 1 — Réseaux et systèmes Projet SLOOP Rapport de recherche n ̊???? — Juillet 1996 — 25 pages Abstract: in this paper, we prove that the wrapped Butterfly digraph ~ WBF(d; n) of degree d and dimensionn contains at least d 1 arc-disjoint Hamilton circuits, answering a conjecture of D. Barth. We also conjecture that ~ WBF(d; ...

In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K = K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem...

Multiple edge-disjoint Hamilton cycles have been obtained in labelled star graphs Stn of degree n-1, using number-theoretic means, as images of a known base 2-labelled Hamilton cycle under label-mapping automorphisms of Stn. However, no optimum bounds for producing such edge-disjoint Hamilton cycles have been given, and no positive or negative results exist on whether Hamilton decompositions ca...

We present a Hamilton cycle in the k-sided pancake network and four combinatorial algorithms to traverse cycle. The network’s vertices are coloured permutations $$\pi = p_1p_2\cdots p_n$$ , where each $$p_i$$ has an associated colour $$\{0,1,\ldots k\,-\,1\}$$ . There is directed edge $$(\pi _1,\pi _2)$$ if _2$$ can be obtained from _1$$ by “flip” of length j, which reverses first j elements in...

Resolving a conjecture of Kühn and Osthus from 2012, we show that p = 1 / n <...

We describe an algorithm which enumerates all Hamilton cycles of a given 3regular n-vertex graph in time O(1.276n), improving on Eppstein’s previous bound. The resulting new upper bound of O(1.276n) for the maximum number of Hamilton cycles in 3-regular n-vertex graphs gets close to the best known lower bound of Ω(1.259n). Our method differs from Eppstein’s in that he considers in each step a n...