Hamilton cycles in hypergraphs below the Dirac threshold

Abstract

We establish a precise characterisation of 4-uniform hypergraphs with minimum codegree close to n/2 which contain a Hamilton 2-cycle. As an immediate corollary we identify the exact Dirac threshold for Hamilton 2-cycles in 4-uniform hypergraphs. Moreover, by derandomising the proof of our characterisation we provide a polynomial-time algorithm which, given a 4-uniform hypergraph H with minimum codegree close to n/2, either finds a Hamilton 2-cycle in H or provides a certificate that no such cycle exists. This surprising result stands in contrast to the graph setting, in which below the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We also consider tight Hamilton cycles in k-uniform hypergraphs H for k ≥ 3, giving a series of reductions to show that it is NP-hard to determine whether a k-uniform hypergraph H with minimum degree δ(H) ≥ 1 2 |V (H)| − O(1) contains a tight Hamilton cycle. It is therefore unlikely that a similar characterisation can be obtained for tight Hamilton cycles.

Hamilton $\ell$-Cycles in Randomly Perturbed Hypergraphs

“…In all other cases the difference is a constant factor. By contrast, the exact value of this threshold (for large n) has only been found in a small number of cases, namely for k = 3, ℓ = 2 by Rödl, Ruciński and Szemerédi [32], for k = 4, ℓ = 2 by Garbe and Mycroft [15], for k = 3 and ℓ = 1 by Czygrinow and Molla [9] and for any k ≥ 3 and ℓ < k/2 by Han and Zhao [17]. For other degree conditions, less still is known; indeed the only cases in which the minimum t-degree threshold for a Hamilton ℓ-cycle is known even asymptotically are the cases k ≥ 3, ℓ < k/2, t = k − 2 (due to Bastos, Mota, Schacht, Schnitzer and Schulenburg [3] with previous results for the case (k, ℓ, t) = (3, 1, 1) due to Buß, Hàn and Schacht [7] and Han and Zhao [18]) and (k, ℓ, t) = (3, 2, 1) (due to Reiher, Rödl, Ruciński, Schacht and Szemerédi [28]).…”

Finding Perfect Matchings in Dense Hypergraphs

“…In this section we prove Theorem 3.4, which finds a perfect matching or a large independent set, thus establishing correctness of Procedure PerfectMatching in the non-extremal case. We adapt (and simplify) the approach of Han [14] via lattice-based absorption and also incorporate a derandomisation argument of Garbe and Mycroft [10] so that we can find a perfect matching (not just test for its existence). 4.1.…”

Minimum degree thresholds for Hamilton $(\ell,k-\ell)$-cycles in $k$-uniform hypergraphs

“…The threshold for all k ≥ 3 and ℓ < k/2 was determined by Han and Zhao [14]. The case k = 4 and ℓ = 2 was determined by Garbe and Mycroft [8]. Recently, the case k ≥ 6, k is even and ℓ = k/2 was determined by Hàn, Han and Zhao [12].…”