Hamilton Cycles in Random Lifts of Directed Graphs

Packing, Counting and Covering Hamilton cycles in random directed graphs

A Hamilton cycle in a digraph is a cycle passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Posá ‘rotationextension’ te...

Counting the Number of Hamilton Cycles in Random Digraphs

We show that there exists a a fully polynomial randomized approximation scheme for counting the number of Hamilton cycles in almost all directed graphs.

Counting Hamilton cycles in sparse random directed graphs

Let D(n, p) be the random directed graph on n vertices where each of the n(n− 1) possible arcs is present independently with probability p. It is known that if p ≥ (log n+ ω(1))/n then D(n, p) typically has a directed Hamilton cycle, and this is best possible. We show that under the same condition, the number of directed Hamilton cycles in D(n, p) is typically n!(p(1 + o(1))) . We also prove a ...

Hamilton cycles in random graphs and directed graphs

Closing Gaps in Problems related to Hamilton Cycles in Random Graphs and Hypergraphs

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...