The number of cliques in graphs of given order and size

TL;DR

This paper establishes tight lower bounds on the minimum number of r-cliques in graphs with fixed vertices and edges, using a combination of combinatorial and analytical methods.

Contribution

It provides new bounds for the minimum number of 3- and 4-cliques in graphs, improving understanding of clique distribution in graphs of given size.

Findings

Derived lower bounds for k_3(n,m) and k_4(n,m) with small approximation error

Used a novel combination of combinatorial and analytical techniques

Results applicable to extremal graph theory problems

Abstract

Let k_r(n,m) denote the minimum number of r-cliques in graphs with n vertices and m edges. For r=3,4 we give a lower bound on k_r(n,m) that approximates k_r(n,m) with an error smaller than n^r/(n^2-2m). The solution is based on a constraint minimization of certain multilinear forms. In our proof, a combinatorial strategy is coupled with extensive analytical arguments.