The minimum vertex degree for an almost-spanning tight cycle in a $3$-uniform hypergraph

Abstract

We prove that any $3$-uniform hypergraph whose minimum vertex degree is at least $\left(\frac{5}{9}+o(1)\right)\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ admits an almost-spanning tight cycle, that is, a tight cycle leaving $o(n)$ vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, Böttcher, Cooley and Mycroft.

10 pages. arXiv admin note: text overlap with arXiv:1411.4957