Resilience for Loose Hamilton Cycles

• DOI: 10.48550/arxiv.2309.14197
• Publication Date: 2023-01-01
• Authors: Alvarado, José D.; Kohayakawa, Yoshi...; Lang, Richard; Mota, Guilherme O.; Stagni, Henrique
• Keywords: 05D15, 05D40; FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); G.2.2

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum $d$-degree threshold for loose Hamiltonicity relative to the random $k$-uniform hypergraph $H_k(n,p)$ coincides with its dense analogue whenever $p \geq n^{- (k-1)/2+o(1)}$. The value of $p$ is approximately tight for $d>(k+1)/2$. This is particularly interesting because the dense threshold itself is not known beyond the cases when $d \geq k-2$.<br/>33 pages, 3 figures<br/>