Related papers: Densities of 3-vertex graphs

Given a random 3-uniform hypergraph $H=H(n,p)$ on $n$ vertices where each triple independently appears with probability $p$, consider the following graph process. We start with the star $G_0$ on the same vertex set, containing all the edges…

Given i.i.d. positive integer valued random variables D_1,...,D_n, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D_1,...,D_n. We give sufficient conditions on the distribution of D_i for…

We consider complete graphs with edge weights and/or node weights taking values in some set. In the first part of this paper, we show that a large number of graphs are completely determined, up to isomorphism, by the distribution of their…

The paper shows that almost every $n$-vertex graph is such that the multiset of its induced subgraphs on $3 \log_2{n}$ vertices is sufficient to determine it up to isomorphism. Therefore, for checking the isomorphism of a pair of $n$-vertex…

We use Razborov's flag algebra method to show an asymptotic upper bound for the maximal induced density $i(\vec P_3)$ of the orgraph $\vec P_3$ in an arbitrary orgraph. A conjecture of Thomass\'e states that $i(\vec P_3)=2/5$. The hitherto…

We equip the edges of a deterministic graph $H$ with independent but not necessarily identically distributed weights and study a generalized version of matchings (i.e. a set of vertex disjoint edges) in $H$ satisfying the property that…

This paper deals with the problem of detecting non-isotropic high-dimensional geometric structure in random graphs. Namely, we study a model of a random geometric graph in which vertices correspond to points generated randomly and…

Let $G$ be a finite, simple, and undirected graph of order $n$ and average degree $d$. Up to terms of smaller order, we characterize the minimal intervals $I$ containing $d$ that are guaranteed to contain some vertex degree. In particular,…

There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph H is its extremal density, as an induced subgraph of G, where |G| ->…

We prove that if T is a tree on n vertices wih maximum degree D and the edge probability p(n) satisfies: np>c*max{D*logn,n^{\epsilon}} for some constant \epsilon>0, then with high probability the random graph G(n,p) contains a copy of T.…

We propose a random bipartite graph with weights assigned to both parts of the vertex sets. Edges are formed independently with probabilities that depend on these weights. This bipartite graph naturally gives rise to a random intersection…

Clustering is well-known to play a prominent role in the description and understanding of complex networks, and a large spectrum of tools and ideas have been introduced to this end. In particular, it has been recognized that the abundance…

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…

A random geometric digraph $G_n$ is constructed by taking $\{X_1,X_2,... X_n\}$ in $\mathbb{R}^2$ independently at random with a common bounded density function. Each vertex $X_i$ is assigned at random a sector $S_i$ of central angle…

It is shown that a simple graph which is embeddable in the real projective plane is minimally 3-rigid if and only if it is (3,6)-tight. Moreover the topologically uncontractible embedded graphs of this type are constructible from one of 8…

The Kohayakawa-Nagle-R\"odl-Schacht conjecture roughly states that every sufficiently large locally $d$-dense graph $G$ on $n$ vertices must contain at least $(1-o(1))d^{|E(H)|}n^{|V(H)|}$ copies of a fixed graph $H$. Despite its important…

A graph is universally $k$-edge-weightable if for every $k$-element set $Q\subset\mathbb{R}$, it admits a proper $Q$-edge weighting. The settled 1-2-3 conjecture implies that for any arithmetic progression $\{a,b,c\}$, every nice regular…

Consider a random multigraph with given vertex degrees constructed by the configuration model. We give a new proof of the fact that, asymptotically for a sequence of such multigraphs with the number of edges tending to infinity, the…

We prove that for every graph $G$, given fixed locations for the vertices of $G$ in $\mathbb{Z}^3$, there is a three-dimensional grid-drawing of $G$ with one bend per edge. The best previous bound was three bends per edge.

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…