On some graph densities in locally dense graphs

TL;DR

This paper explores graph densities in locally dense graphs, proving the Kohayakawa-Nagle-R"odl-Schacht conjecture for new classes of graphs and establishing conditions under which these graphs contain many copies of a fixed subgraph.

Contribution

The paper introduces new classes of graphs satisfying the conjecture, including graphs formed by adding edges to cycles or trees and those constructed by gluing multipartite graphs.

Findings

Adding an edge to a cycle or a tree satisfies the conjecture.

Gluing complete multipartite graphs in a tree-like manner satisfies the conjecture.

Analogous results hold when replacing complete multipartite graphs with odd cycles.

Abstract

The Kohayakawa-Nagle-R\"odl-Schacht conjecture roughly states that every sufficiently large locally $d$-dense graph $G$ on $n$ vertices must contain at least $(1-o(1))d^{|E(H)|}n^{|V(H)|}$ copies of a fixed graph $H$. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the conjecture. Firstly, we prove that adding an edge to a cycle or a tree produces graphs that satisfy the conjecture. Secondly, we prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies the conjecture. We also prove an analogous result with odd cycles replacing complete multipartite graphs.