Consider the random graph process where we start with an empty graph on n vertices, and at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that whp the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph has a directe...

We show that p = √ e n is a sharp threshold for the random graph Gn,p to contain the square of a Hamilton cycle. This improves the previous results of Kühn and Osthus and also Nenadov and Škorić.

An n-partite tournament is an orientation of a complete n-partite graph. An npartite tournament is a tournament, if it contains exactly one vertex in each partite set. Douglas, Proc. London Math. Soc. 21 (1970) 716-730, obtained a characterization of strongly connected tournaments with exactly one Hamilton cycle (i.e., n-cycle). For n ≥, we characterize strongly connected n-partite tournaments ...

Let G be a graph on n t> 3 vertices. Dirac's minimum degree condition is the condition that all vertices of G have degree at least in. This is a well-known sufficient condition for the existence of a Hamilton cycle in G. We give related sufficiency conditions for the existence of a Hamilton cycle or a perfect matching involving a restriction of Dirac's minimum degree condition to certain subset...

We consider how many random edges need to be added to a graph of order n with minimum degree αn in order that it contains the square of a Hamilton cycle w.h.p..

We prove that the sum of two random trees possesses with high probability a perfect matching and the sum of five random trees possesses with high probability a Hamilton cycle. AMS Subject Classification.(1991) 05C80

In an earlier paper the authors constructed a hamilton cycle embedding of Kn,n,n in a nonorientable surface for all n ≥ 1 and then used these embeddings to determine the genus of some large families of graphs. In this two-part series, we extend those results to orientable surfaces for all n 6= 2. In part II, a voltage graph construction is presented for building embeddings of the complete tripa...

Article history: Received 30 April 2008 Available online 25 January 2010

This research develops a polynomial time algorithm for Hamilton Cycle(Path) and proves its correctness. A program is developed according to this algorithm and it works very well. This paper declares the research process, algorithm as well as its proof, and the experiment data. Even only the experiment data is a breakthrough.

Given a simple graph G with a vertex set V and a set of "requirements" {rvw \ v , w E V}, we consider a problem to find a Hamilton cycle minimizing an objective function similar to that in the optimum requirement spanning tree (ORST) problem studied by Hu. And we show that a particular Hamilton cycle C" which is explicitly definable is a solution to the problem when G is complete and { r v w } ...