On graphs that contain exactly k copies of a subgraph, and a related problem in search theory

TL;DR

This paper investigates the maximum number of edges in graphs with exactly k copies of a subgraph F, establishes asymptotic equivalence with Turán numbers, and connects these findings to a search problem involving graph queries.

Contribution

It determines the asymptotic behavior of graphs with exactly k copies of F and provides exact values for specific cases, linking extremal graph theory to search theory.

Findings

xa_k(n,F)=(1+o(1))x(n,F) for any F and k

Exact values of xa_k(n,K_3) and xa_1(n,K_r) for large n

Number of NO answers in queries is at least inom{n}{2}-xa_1(n,F)

Abstract

We study $\mathrm{exa}_k(n,F)$, the largest number of edges in an $n$-vertex graph $G$ that contains exactly $k$ copies of a given subgraph $F$. The case $k=0$ is the Tur\'an number $\mathrm{ex}(n,F)$ that is among the most studied parameters in extremal graph theory. We show that for any $F$ and $k$, $\mathrm{exa}_k(n,F)=(1+o(1))\mathrm{ex}(n,F))$ and determine the exact values of $\mathrm{exa}_k(n,K_3)$ and $\mathrm{exa}_1(n,K_r)$ for $n$ large enough. We also explore a connection to the following well-known problem in search theory. We are given a graph of order $n$ that consists of an unknown copy of $F$ and some isolated vertices. We can ask pairs of vertices as queries, and the answer tells us whether there is an edge between those vertices. Our goal is to describe the graph using as few queries as possible. Aigner and Triesch in 1990 showed that the number of queries needed is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$. Among other results we show that the number of queries that were answered NO is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$.