Minimum vertex degree threshold for loose Hamilton cycles in 3-uniform hypergraphs

We show that for sufficiently large n, every 3-uniform hypergraph on n vertices with minimum vertex degree at least ( n - 1 2 ) - ( ? 3 4 n ? 2 ) + c , where c = 2 if n ? 4 N and c = 1 if n ? 2 N ? 4 N , contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Buis, Hi?n and Schacht who proved the corresponding asymptotical result.

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