Cliques in graphs with bounded minimum degree

TL;DR

This paper investigates the minimum number of r-cliques in graphs with given minimum degree, providing exact evaluations for certain degree ranges and proposing a conjecture for extremal graph structures.

Contribution

It evaluates the function k_r(n, δ) for specific degree bounds and introduces a conjectural construction for extremal graphs under certain conditions.

Findings

Exact values of k_r(n, δ) for δ ≤ 4n/5

A proposed construction for extremal graphs

Conjecture on the characterization of extremal graphs

Abstract

Let $k_r(n,\delta)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $\delta$. We evaluate $k_r(n,\delta)$ for $\delta \leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture to give all extremal graphs (subject to certain conditions on $n$, $\delta$ and $r$).