A k-uniform tight cycle C k s is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd(k, s) = 1 or k/ gcd(k, s) is even. We prove that if s ≥ 2k 2 and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V (H)|, then every vertex is covered by a copy of C k s . The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order of which a tight path wraps around a complete k-partite k-uniform hypergraph, which may be of independent interest.For hypergraphs F and H, a perfect F -tiling in H is a spanning collection of vertex-disjoint copies of F . For k ≥ 3, there are currently only a handful of known F -tiling results when F is k-uniform but not k-partite. If s ≡ 0 mod k, then C k s is not k-partite. Here we prove an F -tiling result for a family of non k-partite k-uniform hypergraphs F . Namely, for s ≥ 5k 2 , every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V (H)| has a perfect C k s -tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

We develop a new framework to study minimum 𝑑degree 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 Erdös-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 Rödl, Ruciński and Szemerédi for 𝑑 = 𝑘 − 1 by determining asymptotically best possible degree conditions for 𝑑 = 𝑘 − 2 and all 𝑘 ⩾ 3. This was proved independently by Polcyn, Reiher, Rödl and Schülke. 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∕ √ 𝑘 − 𝑑 due to Han and Zhao.

We show that 3-graphs on n vertices whose codegree is at least p2{3 `op1qqn can be decomposed into tight cycles and admit Euler tours, subject to the trivial necessary divisibility conditions. We also provide a construction showing that our bounds are best possible up to the op1q term. All together, our results answer in the negative some recent questions of Glock, Joos, Kühn, and Osthus.

Given integers k, j with 1 ≤ j ≤ k − 1, we consider the length of the longest j-tight path in the binomial random k-uniform hypergraph H k (n, p). We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges.In particular, for the supercritical case we introduce the Pathfinder algorithm, a depth-first search algorithm which discovers j-tight paths in a k-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm will find a long j-tight path.

The strong chromatic number χ s (G) of a graph G on n vertices is the least number r with the following property: after adding r⌈n/r⌉−n isolated vertices to G and taking the union with any collection of spanning disjoint copies of K r in the same vertex set, the resulting graph has a proper vertex-colouring with r colours. We show that for every c > 0 and every graph G on n vertices with ∆(G) ≥ cn, χ s (G) ≤ (2 + o(1))∆(G), which is asymptotically best possible.2010 Mathematics subject classification: 05C15, 05C70, 05C35.