Abstract
In the random $k$-uniform hypergraph $H_{n,p;k}$ of order $n$ each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle of order $n$ in which every pair of adjacent edges intersects in a single vertex. We prove that if $p n^{k-1}/\log n$ tends to infinity with $n$ then $$\lim_{\substack{n\to \infty\\ 2(k-1) |n}}\Pr(H_{n,p;k}\text{ contains a loose Hamilton cycle})=1.$$ This is asymptotically best possible.
Highlights
The threshold for the existence of Hamilton cycles in the random graph Gn,p has been known for many years, see, e.g., [1], [3] and [9]
We prove that if pnk−1/ log n tends to infinity with n nl→im∞ Pr(Hn,p;k contains a loose Hamilton cycle) = 1
We say that a k-uniform hypergraph (V, E) is a loose Hamilton cycle if there exists a cyclic ordering of the vertices V such that every edge consists of k consecutive
Summary
The threshold for the existence of Hamilton cycles in the random graph Gn,p has been known for many years, see, e.g., [1], [3] and [9]. We say that a k-uniform hypergraph (V, E) is a loose Hamilton cycle if there exists a cyclic ordering of the vertices V such that every edge consists of k consecutive. (log n)/nk−1 is the asymptotic threshold for the existence of loose Hamilton cycles, at least for n a multiple of 2(k−1). This is because if p ≤ (1−ǫ)(k−1)!(log n)/nk−1 and ǫ > 0 is constant, whp Hn,p;k contains isolated vertices. We cannot apply a second moment calculation directly to the number of Hamilton cycles in Hn,p;k, this does not work
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