Abstract: We prove an analogue of Alon's spectral gap conjecture for random bipartite, biregular graphs. We use the Ihara-Bass formula to connect the non-backtracking spectrum to that of the adjacency matrix, employing the moment method to show there exists a spectral gap for the non-backtracking matrix. A byproduct of our main theorem is that random rectangular zero-one matrices with fixed row and column sums are full-rank with high probability. Finally, we illustrate applications to community detection, coding theory, and deterministic matrix completion.

Abstract: Two-sided bounds are explored for concentration functions and Renyi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities.

Abstract: Considering a natural generalization of the Ruzsa–Szemeredi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing copies of Ks such that any Kr is contained in at most one Ks. We also give bounds for the generalized rainbow Turan problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Meszaros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with copies of Kr such that no Kr is rainbow.

Abstract: Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing, etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, for example the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. On all these problems, the probability of deviating by an r-factor in lower-order terms of the expected time decreases exponentially with r. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Finally, user-friendly specializations of the general drift theorem are given.

Abstract: Let G be a simple graph that is properly edge-coloured with m colours and let \[\mathcal{M} = \{ {M_1},...,{M_m}\} \] be the set of m matchings induced by the colours in G. Suppose that \[m \leqslant n - {n^c}\], where \[c > 9/10\], and every matching in \[\mathcal{M}\] has size n. Then G contains a full rainbow matching, i.e. a matching that contains exactly one edge from Mi for each \[1 \leqslant i \leqslant m\]. This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalization of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs. Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger.

Abstract: The size-Ramsey number of a graph $F$ is the smallest number of edges in a graph $G$ with the Ramsey property for $F$, that is, with the property that any 2-colouring of the edges of $G$ contains a monochromatic copy of $F$ We prove that the size-Ramsey number of the grid graph on $n\times n$ vertices is bounded from above by $n^{3+o(1)}$

Abstract: We prove the Erdős–Sos conjecture for trees with bounded maximum degree and large dense host graphs As a corollary, we obtain an upper bound on the multicolour Ramsey number of large trees whose maximum degree is bounded by a constant

Abstract: We prove a number of results related to a problem of Po-Shen Loh [9], which is equivalent to a problem in Ramsey theory. Let a = (a1, a2, a3) and b = (b1, b2, b3) be two triples of integers. Define a to be 2-less than b if ai < bi for at least two values of i, and define a sequence a1, …, am of triples to be 2-increasing if ar is 2-less than as whenever r < s. Loh asks how long a 2-increasing sequence can be if all the triples take values in {1, 2, …, n}, and gives a log* improvement over the trivial upper bound of n2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n3/2. We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well-known problems in extremal combinatorics, and asking a number of further questions.

Abstract: A k-uniform tight cycle . The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order in 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 -tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

Abstract: In this work we consider three well-studied broadcast protocols: push, pull and push&pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures. We explore in particular the following notion of local resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary may delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. On one hand, pull is robust with respect to all parameters that we consider. On the other hand, push may slow down significantly, even if the adversary may modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, push&pull is robust when no message transmission failures are considered, otherwise it may be slowed down. On the technical side, we develop two novel methods for the analysis of randomized rumour-spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemeredi’s Regularity Lemma.

Abstract: We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry-quasirandom 3-graphs. Our first result asserts that for any fixed real α > 0, cherry-quasirandom 3-graphs of sufficiently large order n having minimum 2-degree at least α(n – 2) have a tight Hamilton cycle. Our second result concerns the minimum 1-degree sufficient for such 3-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every d, α > 0 satisfying d + α > 1, any sufficiently large n-vertex such 3-graph H of density d and minimum 1-degree at least has a tight Hamilton cycle.

Abstract: We give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.

Abstract: Erdős, Gyarfas and Pyber showed that every r-edge-coloured complete graph Kn can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if , then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint monochromatic cycles. This answers a question of Korandi, Mousset, Nenadov, Skoric and Sudakov.

Abstract: To develop an efficient protocol, which involves an exploration of the catalytic potential of both the room temperature and surfactant ionic liquids towards the synthesis of biologically important derivatives of 2-aminothiazole. Specific heat capacity data as a function of temperature for the synthesized 2- aminothiazole derivatives has been advanced by exploring their thermal profiles. The thermal gravimetry analysis and differential scanning calorimetry techniques are used systematically. The present strategy could prove to be useful for researchers working in the field of surfactants and surfactant-based ionic liquids towards their exploration in organic synthesis. In addition to that, the effect of electronic parameters on the melting temperature of the corresponding 2-aminothiazole has been demonstrated with the help of thermal analysis. Specific heat capacity data as a function of temperature for the synthesized 2- aminothiazole derivatives has also been reported. Melting behavior of the synthesized 2-aminothiazole derivatives is to be described on the basis of electronic effects with the help of thermal analysis. Additionally, the specific heat capacity data can be helpful for the chemists, those engaged in chemical modelling as well as docking studies. Furthermore, the data also helps to determine valuable thermodynamic parameters such as entropy and enthalpy.

Abstract: We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number $n$ of vertices on discrete tori and bounded degree trees, of order $\mathcal{O}(n \log \log n)$ on bounded degree expanders, and of order $\mathcal{O}(n (\log \log n)^2)$ on the Erdős-Renyi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy, and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

Abstract: Consider a random $n\times n$ zero-one matrix with "density" $p$, sampled according to one of the following two models: either every entry is independently taken to be one with probability $p$ (the "Bernoulli" model), or each row is independently uniformly sampled from the set of all length-$n$ zero-one vectors with exactly $pn$ ones (the "combinatorial" model). We give simple proofs of the (essentially best-possible) fact that in both models, if $\min(p,1-p)\geq (1+\varepsilon)\log n/n$ for any constant $\varepsilon>0$, then our random matrix is nonsingular with probability $1-o(1)$. In the Bernoulli model this fact was already well-known, but in the combinatorial model this resolves a conjecture of Aigner-Horev and Person.

Abstract: The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the Van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver's inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilized to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g., the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvari, which settled Friedland's lower matching conjecture.

Abstract: It has been conjectured that, for any fixed \[{\text{r}} \geqslant 2\] and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every \[({\text{r}} - 1)\]-colouring of the edges of \[{\text{K}}_{\text{n}}^{\text{r}}\], the complete r-uniform hypergraph on n vertices. In this paper we prove this conjecture.

Abstract: We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c 1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree then every vertex of G is contained in a copy of F in G? We asymptotically determine c 1(n, F) when F is the generalized triangle , and we give close to optimal bounds in the case where F is the tetrahedron (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.

Abstract: We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When , there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when , any proper q-colouring of the boundary of a box of side length can be extended to a proper q-colouring of the entire box. (3) When , the latter holds for any . Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.

Abstract: We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the "winding" technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514-527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.

Abstract: A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rodl, Rucinski and Szemeredi proved that for , every k-graph on n vertices with minimum codegree at least contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is . As a corollary, we obtain a similar estimate on the number of Hamilton -cycles in such k-graphs for all , which makes progress on a question of Ferber, Krivelevich and Sudakov.

Abstract: We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d 1 , …, d n ) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes d j = λn + O(n 1/2+e ) for some λ bounded away from 0 and 1.

Abstract: We prove a ‘resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that whenever d is sufficiently large compared to , a.a.s. the following holds. Let be any subgraph of the random n-vertex d-regular graph with minimum degree at least . Then is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly the condition that d is large cannot be omitted, and secondly the minimum degree bound cannot be improved.

Abstract: A set $S$ of permutations is forcing if for any sequence $\{\Pi_i\}_{i \in \mathbb{N}}$ of permutations where the density $d(\pi,\Pi_i)$ converges to $\frac{1}{|\pi|!}$ for every permutation $\pi \in S$, it holds that $\{\Pi_i\}_{i \in \mathbb{N}}$ is quasirandom. Graham asked whether there exists an integer $k$ such that the set of all permutations of order $k$ is forcing; this has been shown to be true for any $k\ge 4$. In particular, the set of all twenty-four permutations of order $4$ is forcing. We provide the first non-trivial lower bound on the size of a forcing set of permutations: every forcing set of permutations (with arbitrary orders) contains at least four permutations.

Abstract: We prove and generalize a conjecture in arXiv:1610.0474(4) about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$, where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.

Abstract: We consider the component structure of the random digraph $D(n,p)$ inside the critical window $p = n^{-1} + \lambda n^{-4/3}$.We show that the largest component $\mathcal{C}_1$ has size of order $n^{1/3}$ in this range. In particular we give explicit bounds on the tail probabilities of $|\mathcal{C}_1|n^{-1/3}$.