Tight bounds for powers of Hamilton cycles in tournaments

Powers of Hamilton Cycles in Pseudorandom Graphs

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (ε, p, k, l)-pseudorandom if for all disjoint X and Y ⊆ V (G) with |X| ≥ εpkn and |Y | ≥ εpln we have e(X,Y ) = (1± ε)p|X||Y |. We prove that for all β > 0 there is an ε > 0 such that an (ε, p, 1, 2)-pseudorandom graph on n vertices with minim...

A sufficient condition for Hamilton cycles in bipartite tournaments

Tight Hamilton cycles in random hypergraphs

We give an algorithmic proof for the existence of tight Hamilton cycles in a random r-uniform hypergraph with edge probability p = n−1+ε for every ε > 0. This partly answers a question of Dudek and Frieze [Random Structures Algorithms], who used a second moment method to show that tight Hamilton cycles exist even for p = ω(n)/n (r ≥ 3) where ω(n) → ∞ arbitrary slowly, and for p = (e + o(1))/n (...

Tight Hamilton cycles in random uniform hypergraphs

In this paper we show that e/n is the sharp threshold for the existence of tight Hamilton cycles in random k-uniform hypergraphs, for all k ≥ 4. When k = 3 we show that 1/n is an asymptotic threshold. We also determine thresholds for the existence of other types of Hamilton cycles.

Packing Tight Hamilton Cycles in Uniform Hypergraphs

We say that a k-uniform hypergraph C is a Hamilton cycle of type `, for some 1 ≤ ` ≤ k, if there exists a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices and for every pair of consecutive edges Ei−1, Ei in C (in the natural ordering of the edges) we have |Ei−1 \ Ei| = `. We define a class of ( , p)-regular hypergraphs, that includes random hypergraph...