Minimum degree conditions for tight Hamilton cycles

• Published in: Journal of the London Mathematical Society 2022-06, Vol.105 (4), p.2249-2323
• Main Authors: Lang, Richard, Sanhueza‐Matamala, Nicolás
• Format: Article
• Language: English
• ISSN: 0024-6107; 1469-7750
• DOI: 10.1112/jlms.12561

We develop a new framework to study minimum d$d$‐degree conditions in k$k$‐uniform hypergraphs, which guarantee the existence of a tight Hamilton cycle. Our main theoretical result deals with the typical absorption, path cover and connecting arguments for all k$k$ and d$d$ at once, and thus sheds light on the underlying structural problems. Building on this, we show that one can study minimum d$d$‐degree conditions of k$k$‐uniform tight Hamilton cycles by focusing on the inner structure of the neighbourhoods. This reduces the matter to an Erdös–Gallai‐type question for (k−d)$(k-d)$‐uniform hypergraphs, which is of independent interest. Once this framework is established, we can easily derive two new bounds. Firstly, we extend a classic result of Rödl, Ruciński and Szemerédi for d=k−1$d=k-1$ by determining asymptotically best possible degree conditions for d=k−2$d = k-2$ and all k⩾3$k \geqslant 3$. This was proved independently by Polcyn, Reiher, Rödl and Schülke. Secondly, we provide a general upper bound of 1−1/(2(k−d))$1-1/(2(k-d))$ for the tight Hamilton cycle d$d$‐degree threshold in k$k$‐uniform hypergraphs, thus narrowing the gap to the lower bound of 1−1/k−d$1-1/\sqrt {k-d}$ due to Han and Zhao.