Pseudo-random Graphs Research Articles

Hamilton cycles in pseudorandom graphs

Finding general conditions which ensure that a graph is Hamiltonian is a central topic in graph theory. An old and well-known conjecture in the area states that any d-regular n-vertex graph G whose second largest eigenvalue in absolute value λ(G) is at most d/C, for some universal constant C>0, has a Hamilton cycle. In this paper, we obtain two main results which make substantial progress towards this problem. Firstly, we settle this conjecture in full when the degree d is at least a small power of n. Secondly, in the general case we show that λ(G)≤d/C(log⁡n)1/3 implies the existence of a Hamilton cycle, improving the 20-year old bound of d/log1−o(1)⁡n of Krivelevich and Sudakov. We use in a novel way a variety of methods, such as a robust Pósa rotation-extension technique, the Friedman-Pippenger tree embedding with rollbacks and the absorbing method, combined with additional tools and ideas.Our results have several interesting applications. In particular, they imply the currently best-known bounds on the number of generators which guarantee the Hamiltonicity of random Cayley graphs, which is an important partial case of the well known Hamiltonicity conjecture of Lovász. They can also be used to improve a result of Alon and Bourgain on additive patterns in multiplicative subgroups.

Efficient polynomial-time approximation scheme for the genus of dense graphs

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that \(|E(G)|\ge \alpha \, |V(G)|^2\) for some fixed \(\alpha \gt 0\) . While a constant-factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph G of order n and given \(\varepsilon \gt 0\) , returns an integer g such that G has an embedding in a surface of genus g , and this is ɛ-close to a minimum genus embedding in the sense that the minimum genus \(\mathsf {g}(G)\) of G satisfies: \(\mathsf {g}(G)\le g\le (1+\varepsilon)\mathsf {g}(G)\) . The running time of the algorithm is \(O(f(\varepsilon)\,n^2)\) , where \(f(\cdot)\) is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is g . This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time \(O(f_1(\varepsilon)\,n^2)\) . Our algorithms are based on the analysis of minimum genus embeddings of quasirandom graphs. We use a general notion of quasirandom graphs [ 25 ]. We start with a regular partition obtained via an algorithmic version of the Szemerédi Regularity Lemma (due to Frieze and Kannan [ 17 ] and to Fox, Lovász, and Zhao [ 14 , 15 ]). We then partition the input graph into a bounded number of quasirandom subgraphs, which are preselected in such a way that they admit embeddings using as many triangles and quadrangles as faces as possible. Here we provide an ɛ-approximation \(\nu (G)\) for the maximum number of edge-disjoint triangles in G . The value \(\nu (G)\) can be computed by solving a linear program whose size is bounded by certain value \(f_2(\varepsilon)\) depending only on ɛ. After solving the linear program, the genus can be approximated (see Corollary 1.7 ). The proof of this result is long and will be of independent interest in topological graph theory.

Markov processes on quasi-random graphs

We study Markov population processes on large graphs, with the local state transition rates of a single vertex being a linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. An explicit error bound is given and it is 1N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{\\sqrt{N}}$$\\end{document} plus the largest discrepancy of the graph. For Erdős–Rényi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process.

Coprime networks of the composite numbers: Pseudo-randomness and synchronizability

In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the network increases, the network will be connected whenever the largest possible node index n≥49. To investigate how the nodes are connected, we analytically describe that the link density saturates to 6/π2, whereas the average degree increases linearly with slope 6/π2 with the size of the network. To investigate how the neighbors of the nodes are connected to each other, we find the shortest path length will be at most 3 for 49≤n≤288 and it is at most 2 for n≥289. We also derive an analytic expression for the local clustering coefficients of the nodes, which quantifies how close the neighbors of a node to form a triangle. We also provide an expression for the number of r-length labeled cycles, which indicates the existence of a cycle of length at most O(logn). Finally, we show that this graph sequence is actually a sequence of weakly pseudo-random graphs. We numerically verify our observed analytical results. As a possible application, we have observed less synchronizability (the ratio of the largest and smallest positive eigenvalue of the Laplacian matrix is high) as compared to Erdős–Rényi random network and Barabási–Albert network. This unusual observation is consistent with the prolonged transient behaviors of ecological and predator–prey networks which can easily avoid the global synchronization.

Turán problems in pseudorandom graphs

AbstractGiven a graph $F$ , we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$ . We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper bound on the density. In particular, our result implies that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must contain a copy of the Peterson graph, while the previous best result gives the bound $n^{-1/4}$ . Moreover, we conjecture that the exponent $1/3$ in our bound is tight. We also construct the densest known pseudorandom $K_{2,3}$ -free graphs that are also triangle-free. Finally, we give a different proof for the densest known construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer, and Pepe that they have no large clique.

Extremal Numbers and Sidorenko’s Conjecture

Abstract Sidorenko’s conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko’s conjecture does not hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $\textrm{ex}(n,H)$, the maximum number of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $3$-partite $3$-uniform tight cycles.

Geometric structures in pseudo-random graphs

Abstract In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.

Clique factors in pseudorandom graphs

An n -vertex graph is said to to be (p,\beta) -bijumbled if for any vertex sets A,B\subseteq V(G) , we have e(A,B)=p|A|\,|B|\pm \beta \sqrt{|A|\,|B|}. We prove that for any r\in {\mathbb N}_{\ge3} and c>0 there exists an \varepsilon>0 such that any n -vertex (p,\beta) -bijumbled graph with n\in r \mathbb{N} , p>0 , \delta(G)\geq cpn and \beta \leq \varepsilon p^{r-1}n contains a K_{r} -factor. This implies a corresponding result for the stronger pseudorandom notion of (n,d,\lambda) -graphs. For the case of triangle factors, that is, when r=3 , this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition of \beta=o(p^{2}n) actually guarantees that a (p,\beta) -bijumbled graph G contains every graph on n vertices with maximum degree at most 2.

Forcing generalised quasirandom graphs efficiently

AbstractWe study generalised quasirandom graphs whose vertex set consists of$q$parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovász and Sós showed that the structure of such graphs is forced by homomorphism densities of graphs with at most$(10q)^q+q$vertices; subsequently, Lovász refined the argument to show that graphs with$4(2q+3)^8$vertices suffice. Our results imply that the structure of generalised quasirandom graphs with$q\ge 2$parts is forced by homomorphism densities of graphs with at most$4q^2-q$vertices, and, if vertices in distinct parts have distinct degrees, then$2q+1$vertices suffice. The latter improves the bound of$8q-4$due to Spencer.

Seymour's Second Neighborhood Conjecture for orientations of (pseudo)random graphs

Seymour's Second Neighborhood Conjecture (SNC) states that every oriented graph contains a vertex whose second neighborhood is as large as its first neighborhood. We investigate the SNC for orientations of both binomial and pseudo random graphs, verifying the SNC asymptotically almost surely (a.a.s.)1.for all orientations of G(n,p) if limsupn→∞p<1/4; and2.for a uniformly-random orientation of each weakly (p,Anp)-bijumbled graph of order n and density p, where p=Ω(n−1/2) and 1−p=Ω(n−1/6) and A>0 is a universal constant independent of both n and p. We also show that a.a.s. the SNC holds for almost every orientation of G(n,p). More specifically, we prove that a.a.s.1.for all ε>0 and p=p(n) with limsupn→∞p≤2/3−ε, every orientation of G(n,p) with minimum outdegree Ωε(n) satisfies the SNC; and2.for all p=p(n), a random orientation of G(n,p) satisfies the SNC. We remark that either (iii) or (iv) confirms the SNC for almost every oriented graph.

A note on pseudorandom Ramsey graphs

For fixed s \ge 3 , we prove that if optimal K_s -free pseudorandom graphs exist, then the Ramsey number r(s,t) is t^{s-1+o(1)} as t \rightarrow \infty . Our method also improves the best lower bounds for r(C_{\ell},t) obtained by Bohman and Keevash from the random C_{\ell} -free process by polylogarithmic factors for all odd \ell \geq 5 and \ell \in \{6,10\} . For \ell = 4 it matches their lower bound from the C_4 -free process. We also prove, via a different approach, that r(C_5, t)&gt; (1+\nobreak o(1))t^{11/8} and r(C_7, t)&gt; (1+\nobreak o(1))t^{11/9} . These improve the exponent of t in the previous best results and appear to be the first examples of graphs F with cycles for which such an improvement of the exponent for r(F, t) is shown over the bounds given by the random F -free process and random graphs.

Perfectly packing graphs with bounded degeneracy and many leaves

We prove that one can perfectly pack degenerate graphs into complete or dense n-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $$o\left( {{n \over {\log \,n}}} \right)$$ , and in addition Ω(n) of them have at most (1 − Ω(1))n vertices and Ω(n) leaves. This proves Ringel’s conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).

1‐independent percolation on ℤ2×Kn

AbstractA random graph model on a host graph is said to be 1‐independent if for every pair of vertex‐disjoint subsets of , the state of edges (absent or present) in is independent of the state of edges in . For an infinite connected graph , the 1‐independent critical percolation probability is the infimum of the such that every 1‐independent random graph model on in which each edge is present with probability at least almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that tends to a limit in as , and they asked for the value of this limit. We make progress on a related problem by showing that In fact, we show that the equality above remains true if the sequence of complete graphs is replaced by a sequence of weakly pseudorandom graphs on vertices with average degree . We conjecture the answer to Balister and Bollobás's question is also .

Chromatic index of dense quasirandom graphs

Let G be a simple graph with maximum degree Δ(G). A subgraph H of G is overfull if |E(H)|>Δ(G)⌊|V(H)|/2⌋. Chetwynd and Hilton in 1986 conjectured that a graph G on n vertices with Δ(G)>n/3 has chromatic index Δ(G) if and only if G contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, Kühn and Osthus is affirmative.

Sandwiching dense random regular graphs between binomial random graphs

Kim and Vu made the following conjecture (Advances in Mathematics, 2004): if dgg log n, then the random d-regular graph {mathscr {G}}(n,d) can asymptotically almost surely be “sandwiched” between {mathscr {G}}(n,p_1) and {mathscr {G}}(n,p_2) where p_1 and p_2 are both (1+o(1))d/n. They proved this conjecture for log nll dleqslant n^{1/3-o(1)}, with a defect in the sandwiching: {mathscr {G}}(n,d) contains {mathscr {G}}(n,p_1) perfectly, but is not completely contained in {mathscr {G}}(n,p_2). The embedding {mathscr {G}}(n,p_1) subseteq {mathscr {G}}(n,d) was improved by Dudek, Frieze, Ruciński and Šileikis to d=o(n). In this paper, we prove Kim–Vu’s sandwich conjecture, with perfect containment on both sides, for all d where min {d, n-d}gg n/sqrt{log n}. The sandwich theorem allows translation of many results from {mathscr {G}}(n,p) to {mathscr {G}}(n,d) such as Hamiltonicity, the chromatic number, the diameter, etc. It also allows translation of threshold functions of phase transitions from {mathscr {G}}(n,p) to bond percolation of {mathscr {G}}(n,d). In addition to sandwiching regular graphs, our results cover graphs whose degrees are asymptotically equal. The proofs rely on estimates for the probability of small subgraph appearances in a random factor of a pseudorandom graph, which is of independent interest.

New bounds for Ryser’s conjecture and related problems

A Latin square of order n n is an n × n n \times n array filled with n n symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The study of Latin squares goes back more than 200 years to the work of Euler. One of the most famous open problems in this area is a conjecture of Ryser-Brualdi-Stein from 60s which says that every Latin square of order n × n n\times n contains a transversal of order n − 1 n-1 . In this paper we prove the existence of a transversal of order n − O ( log ⁡ n / log ⁡ log ⁡ n ) n-O(\log {n}/\log {\log {n}}) , improving the celebrated bound of n − O ( log 2 ⁡ n ) n-O(\log ^2n) by Hatami and Shor. Our approach (different from that of Hatami-Shor) is quite general and gives several other applications as well. We obtain a new lower bound on a 40-year-old conjecture of Brouwer on the maximum matching in Steiner triple systems, showing that every such system of order n n is guaranteed to have a matching of size n / 3 − O ( log ⁡ n / log ⁡ log ⁡ n ) n/3-O(\log {n}/\log {\log {n}}) . This substantially improves the current best result of Alon, Kim and Spencer which has the error term of order n 1 / 2 + o ( 1 ) n^{1/2+o(1)} . Finally, we also show that O ( n log ⁡ n / log ⁡ log ⁡ n ) O(n\log {n}/\log {\log {n}}) many symbols in Latin arrays suffice to guarantee a full transversal, improving on a previously known bound of n 2 − ε n^{2-\varepsilon } . The proofs combine in a novel way the semi-random method together with the robust expansion properties of edge-coloured pseudorandom graphs to show the existence of a rainbow matching covering all but O ( log ⁡ n / log ⁡ log ⁡ n ) O(\log n/\log {\log {n}}) vertices. All previous results, based on the semi-random method, left uncovered at least Ω ( n α ) \Omega (n^{\alpha }) (for some constant α &gt; 0 \alpha &gt;0 ) vertices.

A clique-free pseudorandom subgraph of the pseudo polarity graph

We provide a new family of Kk-free pseudorandom graphs with edge density Θ(n−1/(k−1)), matching a recent construction due to Bishnoi, Ihringer and Pepe [2]. As in the former result, the idea is to use large subgraphs of polarity graphs, which are defined over a finite field Fq. While their construction required q to be odd, we will give the first construction with q a power of 2.

Hamiltonian cycles above expectation in r-graphs and quasi-random r-graphs

Let Hr(n,p) denote the maximum number of Hamiltonian cycles in an n-vertex r-graph with density p∈(0,1). The expected number of Hamiltonian cycles in the random r-graph model Gr(n,p) is E(n,p)=pn(n−1)!/2 and in the random graph model Gr(n,m) with m=p(nr) it is, in fact, slightly smaller than E(n,p).For graphs, H2(n,p) is proved to be only larger than E(n,p) by a polynomial factor and it is an open problem whether a quasi-random graph with density p can be larger than E(n,p) by a polynomial factor.For hypergraphs (i.e. r≥3) the situation is drastically different. For all r≥3 it is proved that Hr(n,p) is larger than E(n,p) by an exponential factor and, moreover, there are quasi-random r-graphs with density p whose number of Hamiltonian cycles is larger than E(n,p) by an exponential factor.

Independent sets in hypergraphs omitting an intersection

AbstractA k‐uniform hypergraph with n vertices is an ‐omitting system if it has no two edges with intersection size . If in addition it has no two edges with intersection size greater than , then it is an ‐system. Rödl and Šiňajová proved a sharp lower bound for the independence number of ‐systems. We consider the same question for ‐omitting systems. Our proofs use adaptations of the random greedy independent set algorithm, and pseudorandom graphs. We also prove related results where we forbid more than two edges with a prescribed common intersection size leading to some applications in Ramsey theory. For example, we obtain good bounds for the Ramsey number , where F is the k‐uniform Fan. The behavior is quite different than the case which is the classical Ramsey number .

A unified view of graph regularity via matrix decompositions

AbstractWe give a unified proof of algorithmic weak and Szemerédi regularity lemmas for several well‐studied classes of sparse graphs, for which only weak regularity lemmas were previously known. These include core‐dense graphs, low threshold rank graphs, and (a version of) upper regular graphs. More precisely, we define cut pseudorandom graphs, we prove our regularity lemmas for these graphs, and then we show that cut pseudorandomness captures all of the above graph classes as special cases. The core of our approach is an abstracted matrix decomposition, which can be computed by a simple algorithm by Charikar. Using work of Oveis Gharan and Trevisan, it also implies new PTASes for MAX‐CUT, MAX‐BISECTION, MIN‐BISECTION for a significantly expanded class of input graphs. (It is NP Hard to get PTASes for these graphs in general.)