Related papers: On the subgraph query problem

An $n$-vertex $k$-uniform hypergraph $G$ is $(d,\alpha)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have…

We estimate the minimum number of distance queries that is sufficient to reconstruct the binomial random graph $G(n,p)$ with constant diameter with high probability. We get a tight (up to a constant factor) answer for all $p>n^{-1+o(1)}$…

The problem of subgraph counting asks for the number of occurrences of a pattern graph $H$ as a subgraph of a host graph $G$ and is known to be computationally challenging: it is $\#W[1]$-hard even when $H$ is restricted to simple…

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…

Recently, Farnik asked whether the hat guessing number $\text{HG}(G)$ of a graph $G$ could be bounded as a function of its degeneracy $d$, and Bosek, Dudek, Farnik, Grytczuk and Mazur showed that $\text{HG}(G)\ge 2^d$ is possible. We show…

Let $\mathcal{F}$ be a collection of $r$-uniform hypergraphs, and let $0 < p < 1$. It is known that there exists $c = c(p,\mathcal{F})$ such that the probability of a random $r$-graph in $G(n,p)$ not containing an induced subgraph from…

Let $H$ be a fixed $k$-vertex graph with $m$ edges and minimum degree $d >0$. We use the learning graph framework of Belovs to show that the bounded-error quantum query complexity of determining if an $n$-vertex graph contains $H$ as a…

Finding the largest clique is a notoriously hard problem, even on random graphs. It is known that the clique number of a random graph G(n,1/2) is almost surely either k or k+1, where k = 2log n - 2log(log n) - 1. However, a simple greedy…

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$. Erd\H{o}s proved that when $n \geq 6d$, each nonhamiltonian graph $G$ on $n$ vertices with minimum degree…

Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem that has been extensively studied in recent years. We study a hypergraph version of the problem. Let $G^r(n,p)$ denote the…

Two central problems in extremal combinatorics are concerned with estimating the number $ex(n,H)$, the size of the largest $H$-free hypergraph on $n$ vertices, and the number $forb(n,H)$ of $H$-free hypergraph on $n$ vertices. While it is…

We consider a problem introduced by Feige, Gamarnik, Neeman, R\'acz and Tetali [2020], that of finding a large clique in a random graph $G\sim G(n,\frac{1}{2})$, where the graph $G$ is accessible by queries to entries of its adjacency…

The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on $n$ vertices. Under…

Let ${H}=(V, {E})$ be a hypergraph on the vertex set $V$ and edge set ${E}\subseteq 2^V$. We show that number of distinct {\it traces} on any $k-$ subset of $V$, is most $k.{\hat \alpha}(H)$, where ${\hat \alpha}(H)$ is the {\it degeneracy}…

We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We…

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph $G\sim {\mathcal G}(n,p)$ in order to find a subgraph which…

We study the problems of counting the homomorphisms, counting the copies, and counting the induced copies of a $k$-vertex graph $H$ in a $d$-degenerate $n$-vertex graph $G$. Our main result establishes exhaustive and explicit complexity…

A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In…

One of the most basic questions one can ask about a graph $H$ is: how many $H$-free graphs on $n$ vertices are there? For non-bipartite $H$, the answer to this question has been well-understood since 1986, when Erd\H{o}s, Frankl and R\"odl…

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…