Minimum degree conditions for tight Hamilton cycles

Abstract: We develop a new framework to study minimum $d$-degree conditions in $k$-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 $k$ and $d$ at once, and thus sheds light on the underlying structural problems. Building on this, we show that one can study minimum $d$-degree conditions of $k$-uniform tight Hamilton cycles by focusing on the inner structure of the neighbourhoods. This reduces the matter to an Erd\H{o}s--Gallai-type question for $(k-d)$-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 R\"odl, Ruci\'nski and Szemer\'edi for $d=k-1$ by determining asymptotically best possible degree conditions for $d = k-2$ and all $k \ge 3$. This was proved independently by Polcyn, Reiher, R\"odl and Sch\"ulke. Secondly, we provide a general upper bound of $1-1/(2(k-d))$ for the tight Hamilton cycle $d$-degree threshold in $k$-uniform hypergraphs, thus narrowing the gap to the lower bound of $1-1/\sqrt{k-d}$ due to Han and Zhao.