Large tight components and Dirac thresholds for tight Hamilton cycles

Hao Lin; Guanghui Wang; Jiande Zhang; Wenling Zhou

paper

Large tight components and Dirac thresholds for tight Hamilton cycles

arXiv:2610.38560

Abstract

For every odd integer $r \geq 1$ , let $q_{r} = 1 - 2^{- r} (( r - 1 ) /2 r)$ . We show that, for every odd $r \geq 3$ and sufficiently large $n$ , every $n$ -vertex $r$ -uniform hypergraph with at least $q_{r} (r n)$ edges contains a tight component with at least $(1/2 - o (1)) (r n)$ edges. This confirms a conjecture posed by Lang and Sanhueza-Matamala in 2024. As an application, for every odd $r \geq 1$ and every integer $k \geq ⌈ r (3 - 2 q_{r})⌉$ , we prove that the asymptotic minimum $(k - r)$ -degree threshold for a tight Hamilton cycle in a $k$ -uniform hypergraph is $q_{r}$ , which settles, in this range, another conjecture of Lang and Sanhueza-Matamala in 2022 stating that the lower-bound construction due to Han and Zhao is asymptotically optimal. Our proof combines an isoperimetric inequality for Johnson graphs with a Kruskal-Katona-type theorem, which is of independent interest.

arXiv comment20 pages