Counting and packing Hamilton cycles in dense graphs and oriented graphs

Counting and packing Hamilton cycles in dense graphs and oriented graphs

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 extremal results. In particular, we show that every nearly cn-regular oriented graph on n vertices with c > 3/8 contains (cn/e)(1 + o(1)) directed Hamilton cycles. This is an extension of a result of Cuckler, who settle...

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...

Approximately Counting Hamilton Paths and Cycles in Dense Graphs

We describe fully polynomial randomized approximation schemes for the problems of determining the number of Hamilton paths and cycles in an n-vertex graph with minimum degree (g + e)n, for any fixed e > 0. We show that the exact counting problems are #P-complete. We also describe fully polynomial randomized approximation schemes for counting paths and cycles of all sizes in such graphs.

Counting and packing Hamilton `-cycles in dense hypergraphs

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)...

Hamilton cycles in dense vertex-transitive graphs