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arXiv · 1304.1426

Loose Hamilton Cycles in Regular Hypergraphs

Andrzej Dudek·Alan Frieze·Andrzej Ruciński·Matas Šileikis
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Abstract

We establish a relation between two uniform models of random $k$-graphs (for constant $k \ge 3$) on $n$ labeled vertices: $H(n,m)$, the random $k$-graph with exactly $m$ edges, and $H(n,d)$, the random $d$-regular $k$-graph. By extending to $k$-graphs the switching technique of McKay and Wormald, we show that, for some range of $d = d(n)$ and a constant $c > 0$, if $m \sim cnd$, then one can couple $H(n,m)$ and $H(n,d)$ so that the latter contains the former with probability tending to one as $n \to \infty$. In view of known results on the existence of a loose Hamilton cycle in $H(n,m)$, we conclude that $H(n,d)$ contains a loose Hamilton cycle when $\log n = o(d)$ (or just $d \ge C log n$, if $k = 3$) and $d = o(n^{1/2})$.

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Andrzej Dudek, Alan Frieze, Andrzej Ruciński, Matas Šileikis. 2013-04-04. Loose Hamilton Cycles in Regular Hypergraphs. https://doi.org/10.1017/s0963548314000406

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