Perfect graphs of fixed density: counting and homogenous sets

TL;DR

This paper characterizes the asymptotic number of perfect graphs with fixed density and shows that almost all such graphs contain large homogeneous sets, answering a question about the Erdős-Hajnal property for C_5-free graphs.

Contribution

It provides an exact asymptotic enumeration of perfect graphs of fixed density and establishes the presence of large homogeneous sets in C_5-free graphs, resolving a previously open problem.

Findings

Logarithmic density of perfect graphs matches that of C_5-free graphs.

Almost all C_5-free graphs have linear-sized homogeneous sets.

The entropy function describes the asymptotic count based on density.

Abstract

For c in [0,1] let P_n(c) denote the set of n-vertex perfect graphs with density c and C_n(c) the set of n-vertex graphs without induced C_5 and with density c. We show that log|P_n(c)|/binom{n}{2}=log|C_n(c)|/binom{n}{2}=h(c)+o(1) with h(c)=1/2 if 1/4<c<3/4 and h(c)=H(|2c-1|)/2 otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in C_n(c) have homogenous sets of linear size. This answers a question raised by Loebl, Reed, Scott, Thomason, and Thomass\'e [Almost all H-free graphs have the Erd\H{o}s-Hajnal property] in the case of forbidden induced C_5.