Minimum degree conditions for tight Hamilton cycles
Richard Lang, Nicolás Sanhueza‐Matamala
Abstract
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.