Let h be an arbitrary n-cycle in S n . Assume h ' is another n-cycle in S n . Then if σ = (h) 1 − h ' , σ is an element of A n , the alternating group of S n . From elementary group theory, every element of S n can be written as a product of (not necessarily disjoint) 3-cycles. Thus, h ' = hσ 1σ 2 ...σ r where each σ i (i = 1,2,...,r) is a 3-cycle. Defining h i = hσ 1σ 2 ...σ i , we now stipula...

We prove that, for sufficiently large n, the first player can win the strong perfect matching and Hamilton cycle games. For both games, explicit winning strategies of the first player are given. In devising these strategies we make use of the fact that explicit fast winning strategies are known for the corresponding weak games.

We present a new method which enables us to find threshold functions for many properties in random intersection graphs. This method is used to establish sharp threshold functions in random intersection graphs for k–connectivity, perfect matching containment and Hamilton cycle containment.

A k-uniform hypergraph H contains a Hamilton `-cycle, if there is a cyclic ordering of the vertices of H such that the edges of the cycle are segments of length k in this ordering and any two consecutive edges fi, fi+1 share exactly ` vertices. We consider problems about packing and counting Hamilton `-cycles in hypergraphs of large minimum degree. Given a hypergraph H, for a d-subset A ⊆ V (H)...

We use a randomised embedding method to prove that for all α > 0 any sufficiently large oriented graph G with minimum in-degree and outdegree δ+(G), δ−(G) ≥ (3/8+α)|G| contains every possible orientation of a Hamilton cycle. This confirms a conjecture of Häggkvist and Thomason.

A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n → ∞, provided d ≥ 8.

An algorithm is described which constructs a long path containing a selected vertex x in a graph G. In hamiltonian graphs, it often finds a hamilton cycle or path. The algorithm uses crossovers of order k ≤ M , where M is a fixed constant, to build a longer and longer path. The method is based on theoretical methods often used to prove graphs hamiltonian.

The Kneser graph K (n; k) has as vertices the k-subsets of f1;2;:::;ng. Two vertices are adjacent if the corresponding k-subsets are disjoint. It was recently proved by the rst author 2] that Kneser graphs have Hamilton cycles for n 3k. In this note, we give a short proof for the case when k divides n. x 1. Preliminaries. Suppose that n k 1 are integers and let n] := f1; 2; :::; ng. We denote t...