# Triangles in $K_s$-saturated graphs with minimum degree $t$

## TL;DR

This paper determines the minimum number of triangles in certain saturated graphs and explores how the count of larger cliques varies with minimum degree, revealing new extremal graph structures.

## Contribution

It establishes exact triangle counts in $K_4$-saturated graphs with minimum degree 4 and constructs graphs with prescribed clique counts for larger $s$ and $r$, showing diverse behaviors.

## Key findings

- Minimum triangles in $K_4$-saturated graphs with degree 4 is exactly $2n-4$.
- Unique extremal graph achieving this minimum.
- Number of $K_r$ in $K_s$-saturated graphs can be controlled independently of minimum degree.

## Abstract

For $n \geq 15$, we prove that the minimum number of triangles in an $n$-vertex $K_4$-saturated graph with minimum degree 4 is exactly $2n-4$, and that there is a unique extremal graph. This is a triangle version of a result of Alon, Erd\H{o}s, Holzman, and Krivelevich from 1996. Additionally, we show that for any $s > r \geq 3$ and $t \geq 2 (s-2)+1$, there is a $K_s$-saturated $n$-vertex graph with minimum degree $t$ that has $\binom{ s-2}{r-1}2^{r-1} n + c_{s,r,t}$ copies of $K_r$. This shows that unlike the number of edges, the number of $K_r$'s ($r >2$) in a $K_s$-saturated graph is not forced to grow with the minimum degree, except for possibly in lower order terms.