On the maximum number of copies of H in graphs with given size and order

TL;DR

This paper investigates the maximum number of copies of a fixed graph H in graphs with specified size and order, showing asymptotic realization by quasi-cliques at high edge densities and exploring bipartite variants.

Contribution

It establishes asymptotic results for the maximum copies of H in graphs with given vertices and edges, including bipartite cases, advancing extremal graph theory understanding.

Findings

Maximum copies of H are asymptotically achieved by quasi-cliques at high edge density.

Results extend to bipartite host graphs.

Provides bounds and asymptotic behavior for extremal graph configurations.

Abstract

We study the maximum number $ex(n,e,H)$ of copies of a graph $H$ in graphs with given number of vertices and edges. We show that for any fixed graph $H$, $ex(n,e,H)$ is asymptotically realized by the quasi-clique provided that the edge density is sufficiently large. We also investigate a variant of this problem, when the host graph is bipartite.