Bipodal structure in oversaturated random graphs

TL;DR

This paper investigates the typical structure of large graphs with fixed edge and subgraph densities, showing that they are usually bipodal with parameters depending on the subgraph's degree sequence, especially when the subgraph density exceeds Erdős-Rényi levels.

Contribution

It establishes that for most edge densities, graphs constrained by a slightly higher subgraph density are bipodal with analytically varying parameters depending on the subgraph's degree sequence.

Findings

Large constrained graphs are bipodal for most densities.

Parameters depend analytically on densities and the degree sequence of H.

Results hold for all but finitely many edge density values.

Abstract

We study the asymptotics of large simple graphs constrained by the limiting density of edges and the limiting subgraph density of an arbitrary fixed graph $H$. We prove that, for all but finitely many values of the edge density, if the density of $H$ is constrained to be slightly higher than that for the corresponding Erd\H{o}s-R\'enyi graph, the typical large graph is bipodal with parameters varying analytically with the densities. Asymptotically, the parameters depend only on the degree sequence of $H$.