A Hamilton Berge cycle of a hypergraph on n vertices is an alternating sequence (v1, e1, v2, . . . , vn, en) of distinct vertices v1, . . . , vn and distinct hyperedges e1, . . . , en such that {v1, vn} ⊆ en and {vi, vi+1} ⊆ ei for every i ∈ [n − 1]. We prove a Dirac-type theorem for Hamilton Berge cycles in random r-uniform hypergraphs by showing that for every integer r ≥ 3 there exists k = k...

We study the Hamilton cycle Maker-Breaker game, played on the edges of the random graph G(n, p). We prove a conjecture from [13], asserting that the property that Maker is able to win this game, has a sharp threshold at log n n . Our theorem can be considered a game-theoretic strengthening of classical results from the theory of random graphs: not only does G(n, p) almost surely admit a Hamilto...

Moreover, there is a polynomial time algorithm to find a Hamilton cycle in a random graph G ∈ Gn,p w.h.p. for all p > lnn+ln lnn+c(n) n , c(n) → ∞ [BFF85]. In this section, we look at a simpler algorithm [AV77] that works for all p ≥ c lnn n−1 for a sufficiently large constant c. This is a slightly weaker result as the value of c, determined by Chernoff bound arguments used in the algorithm’s a...

Is it possible to label the edges of Kn with distinct integer weights so that every Hamilton cycle has the same total weight? We give a local condition characterizing the labellings that witness this question’s perhaps surprising affirmative answer. More generally, we address the question that arises when “Hamilton cycle” is replaced by “k-factor” for nonnegative integers k. Such edge-labelling...

The middle levels problem is to find a Hamilton cycle in the middle levels, M2k+1, of the Hasse diagram of B2k+1 (the partially ordered set of subsets of a 2k + 1element set ordered by inclusion). Previously, the best known, from [1], was that M2k+1 is Hamiltonian for all positive k through k = 15. In this note we announce that M33 and M35 have Hamilton cycles. The result was achieved by an alg...

Kreweras’ conjecture [1] asserts that any perfect matching of the hypercube Qd, d ≥ 2, can be extended to a Hamilton cycle. We prove this conjecture.