Supersaturation Problem for Color-Critical Graphs

TL;DR

This paper determines the asymptotic behavior of the minimum number of copies of color-critical graphs in graphs with slightly more edges than the extremal number, extending understanding of supersaturation in graph theory.

Contribution

It provides the first asymptotic determination of $h_F(n,q)$ for color-critical graphs when $q=o(n^2)$ and establishes that the threshold ratio $c_1$ is positive for all such graphs.

Findings

Established asymptotic formulas for $h_F(n,q)$ for color-critical graphs.

Proved that $c_1>0$ for every color-critical graph.

Calculated $c_1$ for specific graphs like odd cycles and bipartite graphs with an added edge.

Abstract

The \emph{Tur\'an function} $\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine $h_F(n,q)$, the minimum number of copies of $F$ that a graph with $n$ vertices and $\ex(n,F)+q$ edges can have. We determine $h_F(n,q)$ asymptotically when $F$ is \emph{color-critical} (that is, $F$ contains an edge whose deletion reduces its chromatic number) and $q=o(n^2)$. Determining the exact value of $h_F(n,q)$ seems rather difficult. For example, let $c_1$ be the limit superior of $q/n$ for which the extremal structures are obtained by adding some $q$ edges to a maximum $F$-free graph. The problem of determining $c_1$ for cliques was a well-known question of Erd\H os that was solved only decades later by Lov\'asz and Simonovits. Here we prove that $c_1>0$ for every {color-critical}~$F$. Our approach also allows us to determine $c_1$ for a number of graphs, including odd cycles, cliques with one edge removed, and complete bipartite graphs plus an edge.