Finding tight Hamilton cycles in random hypergraphs faster

In an $r$-uniform hypergraph on $n$ vertices a tight Hamilton cycle consists of $n$ edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of $r$ vertices. We provide a first deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton cycles in random $r$-uniform hypergraphs with edge probability at least $C \log^3n/n$. Our result partially answers a question of Dudek and Frieze [Random Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton cycles exists already for $p=\omega(1/n)$ for $r=3$ and $p=(e + o(1))/n$ for $r\ge 4$ using a second moment argument. Moreover our algorithm is superior to previous results of Allen, B\"ottcher, Kohayakawa and Person [Random Structures & Algorithms 46 (2015), 446-465] and Nenadov and \v{S}kori\'c [arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a randomised polynomial time algorithm working for edge probabilities $p\ge n^{-1+\varepsilon}$, while the algorithm of Nenadov and \v{S}kori\'c is a randomised quasipolynomial time algorithm working for edge probabilities $p\ge C\log^8n/n$.Comment: 17 page

The anti-Ramsey threshold of complete graphs

For graphs $G$ and $H$, let G {\displaystyle\smash{\begin{subarray}{c} \hbox{\tiny\rm rb} \\ \longrightarrow \\ \hbox{\tiny\rm p} \end{subarray}}}H denote the property that for every proper edge-colouring of $G$ there is a rainbow $H$ in $G$. It is known that, for every graph $H$, an asymptotic upper bound for the threshold function $p^{\rm rb}_H=p^{\rm rb}_H(n)$ of this property for the random graph $G(n,p)$ is $n^{-1/m^{(2)}(H)}$, where $m^{(2)}(H)$ denotes the so-called maximum $2$-density of $H$. Extending a result of Nenadov, Person, \v{S}kori\'c, and Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower bound for $p^{\rm rb}_{K_k}$ for $k\geq 5$. Furthermore, we show that $p^{\rm rb}_{K_4} = n^{-7/15}$.Comment: 19 page

Embedding spanning bounded degree graphs in randomly perturbed graphs

We study the model G 8 G(n; p) of randomly perturbed dense graphs, where G is any n-vertex graph with minimum degree at least n and G(n; p) is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model using absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two new results. For every > 0 and C 5, and every n-vertex graph F with maximum degree at most , we show that if p = !(n−2~(+1)) then G 8 G(n; p) with high probability contains a copy of F. The bound used for p here is lower by a log-factor in comparison to the conjectured threshold for the general appearance of such subgraphs in G(n; p) alone, a typical feature of previous results concerning randomly perturbed dense graphs. We also give the rst example of graphs where the appearance threshold in G 8 G(n; p) is lower than the appearance threshold in G(n; p) by substantially more than a log-factor. We prove that, for every k C 2 and > 0, there is some > 0 for which the kth power of a Hamilton cycle with high probability appears in G 8 G(n; p) when p = !(n−1~k−). The appearance threshold of the kth power of a Hamilton cycle in G(n; p) alone is known to be n−1~k, up to a log-term when k = 2, and exactly for k > 2

Finding tight Hamilton cycles in random hypergraphs faster

In an r-uniform hypergraph on n vertices, a tight Hamilton cycle consists of n edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of r vertices. We provide a first deterministic polynomial-time algorithm, which finds a.a.s. tight Hamilton cycles in random r-uniform hypergraphs with edge probability at least C log 3 n/n. Our result partially answers a question of Dudek and Frieze, who proved that tight Hamilton cycles exist already for p = ω(1/n) for r = 3 and p = (e + o(1))/n for r ≽ 4 using a second moment argument. Moreover our algorithm is superior to previous results of Allen, Böttcher, Kohayakawa and Person, and Nenadov and Škorić, in various ways: the algorithm of Allen et al. is a randomized polynomial-time algorithm working for edge probabilities p ≽ n −1+ ε, while the algorithm of Nenadov and Škorić is a randomized quasipolynomial-time algorithm working for edge probabilities p ≽ C log 8 n/n

Cycle factors in randomly perturbed graphs

We study the problem of finding pairwise vertex-disjoint copies of the ω>-vertex cycle Cω>in the randomly perturbed graph model, which is the union of a deterministic n-vertex graph G and the binomial random graph G(n, p). For ω>≥ 3 we prove that asymptotically almost surely G U G(n, p) contains min{δ(G), min{δ(G), [n/l]} pairwise vertex-disjoint cycles Cω>, provided p ≥ C log n/n for C sufficiently large. Moreover, when δ(G) ≥ αn with 0 ≤ α/l and G and is not 'close' to the complete bipartite graph Kαn(1 - α)n, then p ≥ C/n suffices to get the same conclusion. This provides a stability version of our result. In particular, we conclude that p ≥ C/n suffices when α > n/l for finding [n/l] cycles Cω>. Our results are asymptotically optimal. They can be seen as an interpolation between the Johansson-Kahn-Vu Theorem for Cω>-factors and the resolution of the El-Zahar Conjecture for Cω>-factors by Abbasi

Triangles in randomly perturbed graphs

We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any n -vertex graph G satisfying a given minimum degree condition and the binomial random graph G(n,p) . We prove that asymptotically almost surely G∪G(n,p) contains at least min{δ(G),⌊n/3⌋} pairwise vertex-disjoint triangles, provided p≥Clogn/n , where C is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs. We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and 2 -universality