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arXiv · 1712.00186

Tight Hamilton cycles in cherry quasirandom $3$-uniform hypergraphs

Elad Aigner Horev·Gil Levy
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Abstract

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

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Elad Aigner Horev, Gil Levy. 2017-12-01. Tight Hamilton cycles in cherry quasirandom $3$-uniform hypergraphs. https://arxiv.org/abs/1712.00186

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