For every strictly 1-balanced graph F, the random graph G(n,p) gets an F-factor exactly at the sharp threshold where F-isolated vertices vanish, confirming Ruciński's conjecture.
Loose Hamilton Cycles in Random Uniform Hypergraphs
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abstract
In the random hypergraph $H_{n,p;k}$ each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle 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 $$\lim_{\substack{n\to \infty 2(k-1) |n}}\Pr(H_{n,p;k}\ contains\ a\ loose\ Hamilton\ cycle)=1.$$ This is asymptotically best possible.
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Sharp Thresholds for Factors in Random Graphs
For every strictly 1-balanced graph F, the random graph G(n,p) gets an F-factor exactly at the sharp threshold where F-isolated vertices vanish, confirming Ruciński's conjecture.