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

Abstract

We show that for sufficiently large n, every 3-uniform hypergraph on n vertices with minimum vertex degree at least (n−12)−(⌊34n⌋2)+c, where c=2 if n∈4N and c=1 if n∈2N∖4N, contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Buß, Hàn and Schacht who proved the corresponding asymptotical result.