Due to a calculation error, the constant in the main theorem is not $80 \sqrt{k\log k}$ but $80\sqrt{k} \log k$. The error was discovered by Xizhi Liu.For a new discussion on the limit of the method used in this paper see the revised arXiv version of the paper.

A subsetCof edges in ak-uniform hypergraphHis aloose Hamilton cycleifCcovers all the vertices ofHand there exists a cyclic ordering of these vertices such that the edges inCare segments of that order and such that every two consecutive edges share exactly one vertex. The binomial randomk-uniform hypergraphHkn,phas vertex set [n] and an edge setEobtained by adding eachk-tuplee∈ ($\binom{[n]}{k}$) toEwith probabilityp, independently at random.Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all buto(|E|) edges, referred to as thepacking problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle inHkn,pis$p=\Theta\biggl(\frac{\log n}{n^{k-1}}\biggr),$the best known bounds for the packing problem are aroundp= polylog(n)/n. Here we make substantial progress and prove the following asymptotically (up to a polylog(n) factor) best possible result: forp≥ logCn/nk−1, a randomk-uniform hypergraphHkn,pwith high probability contains$N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$edge-disjoint loose Hamilton cycles.Our proof utilizes and modifies the idea of ‘online sprinkling’ recently introduced by Vu and the first author.

Let G be a finite D-quasirandom group and A ⊂ Gk a δ-dense subset. Then the density of the set of side lengths g of corners $$ \{(a_{1},\dotsc,a_{k}),(ga_{1},a_{2},\dotsc,a_{k}),\dotsc,(ga_{1},\dotsc,ga_{k})\} \subset A $$ converges to 1 as D → ∞.

We identify the asymptotic probability of a configuration model CMn(d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition withkparts if there exists an ε-regular partition withkparts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

We say a graph is (Qn,Qm)-saturatedif it is a maximalQm-free subgraph of then-dimensional hypercubeQn. A graph is said to be (Qn,Qm)-semi-saturatedif it is a subgraph ofQnand adding any edge forms a new copy ofQm. The minimum number of edges a (Qn,Qm)-saturated graph (respectively (Qn,Qm)-semi-saturated graph) can have is denoted by sat(Qn,Qm) (respectivelys-sat(Qn,Qm)). We prove that$$ \begin{linenomath} \lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0, \end{linenomath}$$for fixedm, disproving a conjecture of Santolupo that, whenm=2, this limit is 1/4. Further, we show by a different method that sat(Qn,Q2)=O(2n), and thats-sat(Qn,Qm)=O(2n), for fixedm. We also prove the lower bound$$ \begin{linenomath} \ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n, \end{linenomath}$$thus determining sat(Qn,Q2) to within a constant factor, and discuss some further questions.

A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs.

We study the lower tail large deviation problem for subgraph counts in a random graph. LetXHdenote the number of copies ofHin an Erdős–Rényi random graph$\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability$\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$for fixed 0 < δ < 1.Thanks to the results of Chatterjee, Dembo and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least forp≥n−αH(and conjecturally for a larger range ofp). We study this variational problem and provide a partial characterization of the so-called ‘replica symmetric’ phase. Informally, our main result says that for everyH, and 0 < δ < δHfor some δH> 0, asp→ 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartiteHand δ close to 1.

It is proved that the median eigenvalues of every connected bipartite graph G of maximum degree at most three belong to the interval [−1, 1] with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$. Moreover, if G is not isomorphic to the Heawood graph, then a positive fraction of its median eigenvalues lie in the interval [−1, 1]. This surprising result has been motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry.

We show that certain topologically defined uniform spanning tree probabilities for graphs embedded in an annulus can be computed as linear combinations of Pfaffians of matrices involving the line-bundle Green's function, where the coefficients count cover-inclusive Dyck tilings of skew Young diagrams.