Turán's theorem for pseudo-random graphs

The Grothendieck constant of a graph G = (V, E) is the least constant K such that for every matrix A : V × V → R:

2015

This thesis provides an introduction to the fundamentals of random graph theory. The study starts introduces the two fundamental building blocks of random graph theory, namely discrete probability and graph theory. The study starts by introducing relevant concepts probability commonly used in random graph theory-these include concentration inequalities such as Chebyshev's inequality and Chernoff's inequality. Moreover we proceed by introducing central concepts in graph theory, which will underpin the later discussion. In particular we provide results such as Mycielski's construction of a family of triangle-free graphs with high chromatic number and results in Ramsey theory. Next we introduce the concept of a random graph and present two of the most famous proofs in graph theory using the theory random graphs. These include the proof of the fact that there are graphs with arbitrarily high girth and chromatic number, and a bound on the Ramsey number R(k, k). Finally we conclude by introducing the notion of a threshold function for a monotone graph property and we present proofs for the threhold functions of certain properties.

Electronic Journal of Combinatorics, 2010

Let F n,tr(n) denote the family of all graphs on n vertices and t r (n) edges, where t r (n) is the number of edges in the Turán's graph T r (n)-the complete r-partite graph on n vertices with partition sizes as equal as possible. For a graph G and a positive integer λ, let P G (λ) denote the number of proper vertex colorings of G with at most λ colors, and let f (n, t r (n), λ) = max{P G (λ) : G ∈ F n,tr(n) }. We prove that for all n ≥ r ≥ 2, f (n, t r (n), r + 1) = P Tr(n) (r + 1) and that T r (n) is the only extremal graph.

Combinatorica, 2011

The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the threshold for the appearance of a k-regular subgraph in the Erdős-Rényi random graph model G(n, p) is at most the threshold for the appearance of a nonempty (k + 2)-core. In particular, this pins down the point of appearance of a k-regular subgraph to a window for p of width roughly 2/n for large n and moderately large k. The result is proved by using Tutte's necessary and sufficient condition for a graph to have a k-factor.

SIAM Journal on Discrete Mathematics, 2015

The expansion G + of a graph G is the 3-uniform hypergraph obtained from G by enlarging each edge of G with a new vertex disjoint from V (G) such that distinct edges are enlarged by distinct vertices. Let ex 3 (n, F) denote the maximum number of edges in a 3-uniform hypergraph with n vertices not containing any copy of a 3-uniform hypergraph F. The study of ex 3 (n, G +) includes some well-researched problems, including the case that F consists of k disjoint edges [6], G is a triangle [5, 9, 18], G is a path or cycle [12, 13], and G is a tree [7, 8, 10, 11, 14]. In this paper we initiate a broader study of the behavior of ex 3 (n, G +). Specifically, we show ex 3 (n, K + s,t) = Θ(n 3−3/s) whenever t > (s − 1)! and s ≥ 3. One of the main open problems is to determine for which graphs G the quantity ex 3 (n, G +) is quadratic in n. We show that this occurs when G is any bipartite graph with Turán number o(n ϕ) where ϕ = 1+ √ 5 2 , and in particular, this shows ex 3 (n, G +) = O(n 2) when G is the three-dimensional cube graph. 1 Introduction An r-uniform hypergraph F , or simply r-graph, is a family of r-element subsets of a finite set. We associate an r-graph F with its edge set and call its vertex set V (F). Given an r-graph F , let ex r (n, F) denote the maximum number of edges in an r-graph on n vertices that does not contain F. The expansion of a graph G is the 3-graph G + with edge set {e ∪ {v e } : e ∈ G} where v e are distinct vertices not in V (G). By definition, the expansion of G has exactly |G| edges. Note that Füredi and Jiang [10, 11] used a notion of expansion to r-graphs for general r, but this paper considers only 3-graphs. Expansions include many important hypergraphs who extremal functions have been investigated, for instance the celebrated Erdős-Ko-Rado Theorem [6] for 3-graphs is the case of expansions of a matching. A well-known result is that ex 3 (n, K + 3) = n−1 2 [5, 9, 18]. If a graph is not 3-colorable

Electronic Notes in Discrete Mathematics, 2013

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and Γ the generalized Turán density π F pΓq denotes the density of a maximum subgraph of Γ, which contains no copy of F. Extending classical Turán type results for odd cycles, we show that π F pΓq " 1{2 provided F is an odd cycle and Γ is a sufficiently pseudorandom graph. In particular, for pn, d, λq-graphs Γ, i.e., n-vertex, d-regular graphs with all non-trivial eigenvalues in the interval r´λ, λs, our result holds for odd cycles of length , provided λ ´2 ! d ´1 n logpnq´p ´2qp ´3q. Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szabó, and Vu, who addressed the case when F is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free pn, d, λq-graphs) shows that our assumption on Γ is best possible up to the polylog-factor for every odd ě 5.

Journal of Combinatorial Theory, Series A

Let F be a family of 3-uniform linear hypergraphs. The linear Turán number of F is the maximum possible number of edges in a 3-uniform linear hypergraph on n vertices which contains no member of F as a subhypergraph. In this paper we show that the linear Turán number of the five cycle C 5 (in the Berge sense) is 1 3 √ 3 n 3/2 asymptotically. We also show that the linear Turán number of the four cycle C 4 and {C 3 , C 4 } are equal asmptotically, which is a strengthening of a theorem of Lazebnik and Verstraëte [17]. We establish a connection between the linear Turán number of the linear cycle of length 2k + 1 and the extremal number of edges in a graph of girth more than 2k − 2. Combining our result and a theorem of Collier-Cartaino, Graber and Jiang [8], we obtain that the linear Turán number of the linear cycle of length 2k + 1 is Θ(n 1+ 1 k) for k = 2, 3, 4, 6.

2021

For an r-uniform hypergraph H and a family of r-uniform hypergraphs F , the relative Turán number ex(H,F) is the maximum number of edges in an F-free subgraph of H. In this paper we give lower bounds on ex(H,F) for certain families of hypergraph cycles F such as Berge cycles and loose cycles. In particular, if C3 ` denotes the set of all 3-uniform Berge `-cycles and H is a 3-uniform hypergraph with maximum degree ∆, we prove ex(H, C 4) ≥ ∆−3/4−o(1)e(H), ex(H, C 5) ≥ ∆−3/4−o(1)e(H), and these bounds are tight up to the o(1) term.

Combinatorics, Probability and Computing, 2017

We investigate the asymptotic version of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph $\mathcal{H}$ k (n, p). For 2⩽k(n) ⩽ n/2, let $N=\binom{n}k$ and $D=\binom{n-k}k$ . We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of $\mathcal{H}$ has size $$(1+o(1))p\ffrac kn N$$ for any $$p\gg \ffrac nk\ln^2\biggl(\ffrac nk\biggr)D^{-1}.$$ This lower bound on p is asymptotically best possible for k = Θ(n). For this range of k and p, we are able to show stability as well. A different behaviour occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D −1 ≪ p ⩽ (n/k)1−ϵ D −1, the largest intersecting subhypergraph of $\mathcal{H}$ k (n, p) has size Θ(ln(pD)ND −1), provided that $k \gg \sqrt{n \ln n}$ . Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}$ k , for essentially all values of p...

Journal of Graph Theory, 2013

Let T 2 (n) denote Turán's graph-the complete 2-partite graph on n vertices with partition sizes as equal as possible. We show that for all n ≥ 4, the graph T 2 (n) has more proper vertex colorings in at most 4 colors than any other graph with the same number of vertices and edges. C 2013

Electronic Notes in Discrete Mathematics, 2011

For the Erdős-Rényi random graph G n,p , we give a precise asymptotic formula for the sizeα t (G n,p) of a largest vertex subset in G n,p that induces a subgraph with average degree at most t, provided that p = p(n) is not too small and t = t(n) is not too large. In the case of fixed t and p, we find that this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution.

ArXiv, 2019

For fixed $s \ge 3$, we prove that if optimal $K_s$-free pseudorandom graphs exist, then the Ramsey number $r(s,t) = t^{s-1+o(1)}$ 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 $r(C_5, t)> (1+o(1))t^{11/8}$ and $r(C_7, t)> (1+o(1))t^{11/9}$. 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.

Electronic Notes in Discrete Mathematics, 2015

A graph G is said to be H(n, ∆)-universal if it contains every graph on at most n vertices with maximum degree at most ∆. It is known that for any ε > 0 and any natural number ∆ there exists c > 0 such that the random graph G(n, p) is asymptotically almost surely H((1 − ε)n, ∆)universal for p ≥ c(log n/n) 1/∆. Bypassing this natural boundary, we show that for ∆ ≥ 3 the same conclusion holds when p ≫ n − 1 ∆−1 log 5 n.

The inductive dimension dim(G) of a finite undirected graph G is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the empty graph has dimension −1. We look at the distribution of the random variable dim on the Erdös-Rényi probability space G(n, p), where each of the n(n − 1)/2 edges appears independently with probability 0 ≤ p ≤ 1. We show that the average dimension dn(p) = Ep,n[dim] is a computable polynomial of degree n(n − 1)/2 in p. The explicit formulas allow experimentally to explore limiting laws for the dimension of large graphs. In this context of random graph geometry, we mention explicit formulas for the expectation Ep,n[χ] of the Euler characteristic χ, considered as a random variable on G(n, p). We look experimentally at the statistics of curvature K(v) and local dimension dim(v) = 1 + dim(S(v)) which satisfy χ(G) = v∈V K(v) and dim(G) = 1 |V | v∈V dim(v). We also look at the signature functions f (p) = Ep[dim], g(p) = Ep[χ] and matrix values functions Av,w(p) = Covp[dim(v), dim(w)], Bv,w(p) = Cov[K(v), K(w)] on the probability space G(p) of all subgraphs of a host graph G = (V, E) with the same vertex set V , where each edge is turned on with probability p.

Discrete Mathematics, 2004

We show the number of triangles of G n,1/2 is almost uniformly distributed among residue classes modulo q, where q is a prime number bounded by (log n). This implies a consequence of a conjecture of Bollobás, Pebody and Riordan (that almost every random graph G n,1/2 is uniquely determined by its Tutte polynomial): almost every pair of independently chosen random graphs G n,1/2 has different Tutte polynomials.