On a Conjecture of Nagy on Extremal Densities

TL;DR

This paper disproves Nagy's conjecture that extremal graph densities are always achieved by quasi-star or quasi-clique structures, providing counterexamples and proposing new conjectures with supporting evidence.

Contribution

The authors disprove Nagy's conjecture by constructing counterexamples and introduce new conjectures on extremal densities in large graphs.

Findings

Counterexamples with 6 vertices and 6 edges.

Nagy's conjecture is false for infinitely many graphs.

Proposed new conjectures with supporting evidence.

Abstract

We disprove a conjecture of Nagy on the maximum number of copies N(G,H) of a fixed graph G in a large graph H with prescribed edge density. Nagy conjectured that for all G, the quantity N(G,H) is asymptotically maximised by either a quasi-star or a quasi-clique. We show this is false for infinitely many graphs, the smallest of which has 6 vertices and 6 edges. We also propose some new conjectures for the behaviour of N(G,H), and present some evidence for them.