Packing Hamilton Cycles in Random and Pseudo-Random Hypergraphs
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We say that a k -uniform hypergraph C is a Hamilton cycle of type ℓ, for some 1 ≤ ℓ ≤ k, if there exists a cyclic ordering of the vertices of Csuch 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| = ℓ. We prove that for k/2 < ℓ ≤ k, with high probability almost all edges of the random k -uniform hypergraphH(n,p,k) with p(n) ≫ log 2n/n can be decomposed into edge-disjoint type ℓ Hamilton cycles. A slightly weaker result is given for ℓ = k/2. We also provide sufficient conditions for decomposing almost all edges of a pseudo-random k -uniform hypergraph into type ℓ Hamilton cycles, for k/2 ≤ ℓ ≤ k. For the case ℓ = k these results show that almost all edges of corresponding random and pseudo-random hypergraphs can be packed with disjoint perfect matchings.