The ACO Seminar (2012-2013)

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several new extremal results, and also to derive new and conceptually simple(r) proofs of some known results in this area.

In particular, we show that every nearly cn-regular oriented graph on n vertices with c > 3/8 contains (cn/e)^n * (1+o(1))^n directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments.

We also prove that every graph G on n vertices of minimum degree at least (1/2+\epsilon)n contains at least (1-\epsilon)reg_{even}(G)/2 edge-disjoint Hamilton cycles, where reg_{even}(G) is the maximum even degree of a spanning regular subgraph of G. This establishes an approximate version of a conjecture of Kuhn, Lapinskas and Osthus.

A joint work with Asaf Ferber (Tel Aviv U.) and Benny Sudakov (UCLA).