Tight co-degree condition for the existence of loose Hamilton cycles in 3-graphs

TL;DR

This paper proves that a minimum co-degree of exactly n/4 in 3-graphs guarantees the existence of loose Hamilton cycles, improving previous bounds and establishing a tight condition.

Contribution

It establishes the exact minimum co-degree threshold of n/4 for loose Hamilton cycles in 3-graphs, confirming the bound is tight.

Findings

Minimum co-degree n/4 suffices for loose Hamilton cycles

The result improves previous bounds from (1/4 + o(1))n

The threshold n/4 is shown to be tight

Abstract

In 2006, K\"{u}hn and Osthus showed that if a 3-graph H on n vertices has minimum co-degree at least (1/4 +o(1))n and n is even then H has a loose Hamilton cycle. In this paper, we prove that the minimum co-degree of n/4 suffices. The result is tight.