Localized Codegree Conditions for Tight Hamilton Cycles in 3-Uniform Hypergraphs

Abstract

We study sufficient conditions for the existence of Hamilton cycles in uniformly dense 3-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles, and Aigner-Horev and Levy considered them for tight Hamilton cycles for a fairly strong notion of uniformly dense hypergraphs. We focus on tight cycles and obtain optimal results for a weaker notion of uniformly dense hypergraphs. We show that if an n-vertex 3-uniform hypergraph H = (V, E) has the property that for any set of vertices X and for any collection P of pairs of vertices, the number of hyperedges composed by a pair belonging to P and one vertex from X is at least (1/4 +o(1))vertical bar X vertical bar vertical bar P vertical bar- o(vertical bar V vertical bar(3)) and H has minimum vertex degree at least Omega(vertical bar V vertical bar(2)), then H contains a tight Hamilton cycle. A probabilistic construction shows that the constant 1/4 is optimal in this context.