Related papers: Upper tails for triangles

In this paper, we prove a local limit theorem for the distribution of the number of triangles in the Erdos-Renyi random graph $G(n,p)$, where $p \in (0,1)$ is a fixed constant. Our proof is based on bounding the characteristic function…

Given $r$-uniform hypergraphs $G$ and $H$ the Tur\'an number $\rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the Erd\H{o}s-R\'enyi random…

We expand Conlon's random algebraic construction to show that for any odd number $k \geq 3$ exists a natural number $c_k$ (the same as Conlon's) such that $\operatorname{ex}(n^a,n,\theta_{k,c_k}) = \Omega_{k,a}((n^{1 + a})^{\frac{k +…

We prove that every graph $G$ for which $\omega(G) \geq 3/4(\Delta(G) + 1)$, has an independent set $I$ such that $\omega(G - I) < \omega(G)$. It follows that a minimum counterexample $G$ to Reed's conjecture satisfies $\omega(G) <…

Reed conjectured that for every graph, $\chi \leq \left \lceil \frac{\Delta + \omega + 1}{2} \right \rceil$ holds, where $\chi$, $\omega$ and $\Delta$ denote the chromatic number, clique number and maximum degree of the graph, respectively.…

A Gallai $k$-colouring of a graph $G$ is a colouring of $E(G)$ with $k$ colours that induces no rainbow triangles, that is, a triangle with edges of 3 different colours. We give a first step towards estimating the number of Gallai…

Considering regular graphs with every edge in a triangle we prove lower bounds for the number of triangles in such graphs. For r-regular graphs with r <= 5 we exhibit families of graphs with exactly that number of triangles and then…

We consider an Erd\H{o}s-R\'{e}nyi graph $\mathbb{G}(n,p)$ on $n$ vertices with edge probability $p$ such that \[ \sqrt{\frac{\log n}{\log \log n}} \ll np \le n^{1/2-o(1)}, \label{eq:abs} \tag{$\dagger$} \] and derive the upper tail large…

We establish upper and lower bounds with matching leading terms for tails of weighted sums of two-sided exponential random variables. This extends Janson's recent results for one-sided exponentials.

What does an Erdos-Renyi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and…

Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…

A celebrated conjecture of Tuza states that in any finite graph the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. For an $r$-uniform hypergraph ($r$-graph) $G$, let…

For a fixed integer $r\geqslant 3$, let $\mathbb{H}_r(n,p)$ be a random $r$-uniform hypergraph on the vertex set $[n]$, where each $r$-set is an edge randomly and independently with probability $p$. The random $r$-generalized triadic…

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…

The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers $s,t$ and an $n$-vertex graph $G$ with $\lfloor n^2/4 \rfloor +t$ edges and triangle covering number $s$, we determine…

Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property.…

A $d$-regular graph on $n$ nodes has at most $T_{\max} = \frac{n}{3} \tbinom{d}{2}$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular graph has at least $c \cdot T_{\max}$ triangles, and…

We consider the Erd\H{o}s-R\'enyi evolution of random graphs, where a new uniformly distributed edge is added to the graph in every step. For every fixed $d\ge 1$, we show that with high probability, the graph becomes rigid in $\mathbb R^d$…

For a graph $G$, denote by $t(G)$ (resp. $b(G)$) the maximum size of a triangle-free (resp. bipartite) subgraph of $G$. Of course $t(G) \geq b(G)$ for any $G$, and a classic result of Mantel from 1907 (the first case of Tur\'an's Theorem)…

We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this…