Related papers: Counting cliques in a random graph

For $r,n\ge2$, an ordered $r$-uniform matching of size $n$ is an $r$-uniform hypergraph on a linearly ordered vertex set $V$, with $|V|=rn$, consisting of $n$ pairwise disjoint edges. There are $\tfrac12\binom{2r}r$ different ways two edges…

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…

We consider random simple temporal graphs in which every edge of the complete graph $K_n$ appears once within the time interval [0,1] independently and uniformly at random. Our main result is a sharp threshold on the size of any maximum…

We prove that there exists a positive constant \epsilon such that if \log n / n \le p \le n^{-1+\epsilon}, then asymptotically almost surely the random graph G ~ G(n,p) contains a collection of \lfloor \delta(G)/2 \rfloor edge-disjoint…

In this paper we establish asymptotics (as the size of the graph grows to infinity) for the expected number of cliques in the Chung--Lu inhomogeneous random graph model in which vertices are assigned independent weights which have tail…

We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of ${\mathcal G}(n,p)$ in order to typically find a subgraph…

A random graph order is a partial order achieved by independently sprinkling relations on a vertex set (each with probability $p$) and adding relations to satisfy the requirement of transitivity. A \textit{post} is an element in a partially…

We prove that if G is an (n,d,lambda)-graph (a d-regular graph on n vertices, all of whose non-trivial eigenvalues are at most lambda) and the following conditions are satisfied: 1. d/lambda >= (log n)^{1+epsilon} for some constant…

For any given graph $H$, we are interested in $p_\mathrm{crit}(H)$, the minimal $p$ such that the Erd\H{o}s-R\'enyi graph $G(n,p)$ contains a copy of $H$ with probability at least $1/2$. Kahn and Kalai (2007) conjectured that…

Given any integer d >= 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k then the…

Maximal clique enumeration appears in various real-world networks, such as social networks and protein-protein interaction networks for different applications. For general graph inputs, the number of maximal cliques can be up to…

In this paper we investigate geometric properties of graphs generated by a preferential attachment random graph model with edge-steps. More precisely, at each time $t\in\mathbb{N}$, with probability $p$ a new vertex is added to the graph (a…

We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give…

Let $\Gamma(n,k)$ be the set of $2$-connected $n$-vertex graphs containing an edge that is not on any cycle of length at least $k+1.$ Let $g_s(n,k)$ denote the maximum number of $s$-cliques in a graph in $\Gamma(n,k).$ Recently, Ji and Ye…

We study how many copies of a graph $F$ that another graph $G$ with a given number of cliques is guaranteed to have. For example, one of our main results states that for all $t\ge 2$, if $G$ is an $n$ vertex graph with $kn^{3/2}$ triangles…

We give an explicit extension of Spencer's result on the biplanar crossing number of the Erdos-Renyi random graph $G(n,p)$. In particular, we show that the k-planar crossing number of $G(n,p)$ is almost surely $\Omega((n^2p)^2)$. Along the…

We study the maximum number of $r$-vertex cliques in $(r-1)$-uniform hypergraphs not containing complete $r$-partite hypergraphs $K_r^{(r-1)}(a_1, \dots, a_r)$. By using the hypergraph removal lemma, we show that this maximum is $o( n^{r -…

We determine the Ramsey number of a connected clique matching. That is, we show that if $G$ is a $2$-edge-coloured complete graph on $(r^2 - r - 1)n - r + 1$ vertices, then there is a monochromatic connected subgraph containing $n$ disjoint…

Given d \in (0,infty) let k_d be the smallest integer k such that d < 2k\log k. We prove that the chromatic number of a random graph G(n,d/n) is either k_d or k_d+1 almost surely.