Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6423687 | Electronic Notes in Discrete Mathematics | 2016 | 6 Pages |
A Hamilton Berge cycle of a hypergraph on n vertices is an alternating sequence (v1,e1,v2,â¦,vn,en) of distinct vertices v1,â¦,vn and distinct hyperedges e1,â¦,en such that {v1,vn}âen and {vi,vi+1}âei for every iâ[nâ1]. We prove a Dirac-type theorem for Hamilton Berge cycles in random r-uniform hypergraphs by showing that for every integer râ¥3 there exists k=k(r) such that for every γ>0 and pâ¥logk(r)â¡(n)nrâ1 asymptotically almost surely every spanning subhypergraph HâH(r)(n,p) with minimum vertex degree δ1(H)â¥(12râ1+γ)p(nâ1râ1) contains a Hamilton Berge cycle. The minimum degree condition is asymptotically tight and the bound on p is optimal up to possibly the logarithmic factor. As a corollary this gives a new upper bound on the threshold of H(r)(n,p) with respect to Berge Hamiltonicity.