Tight Hamilton cycles in cherry-quasirandom 3-uniform hypergraphs

Source abstract

Abstract We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry -quasirandom 3-graphs. Our first result asserts that for any fixed real α > 0, cherry-quasirandom 3-graphs of sufficiently large order n having minimum 2-degree at least α ( n – 2) have a tight Hamilton cycle. Our second result concerns the minimum 1-degree sufficient for such 3-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every d , α > 0 satisfying d + α > 1, any sufficiently large n -vertex such 3-graph H of density d and minimum 1-degree at least $\alpha \left({\matrix{{n - 1} \cr 2 \cr } } \right)$ has a tight Hamilton cycle.