# Minimizing the numbers of cliques and cycles of fixed size in an $F$-saturated graph

## TL;DR

This paper advances the understanding of $F$-saturated graphs by determining minimal counts of cliques and cycles, classifying extremal graphs, and addressing longstanding conjectures about saturation limits.

## Contribution

It proves the minimal number of cliques and cycles in $K_s$-saturated graphs for large $n$, classifies extremal cases, and shows the non-existence of saturation limits for infinitely many graph families.

## Key findings

- Confirmed a conjecture on minimal cliques in $K_s$-saturated graphs.
- Classified extremal graphs for cycles of most lengths.
- Constructed graph families where saturation limits do not exist.

## Abstract

This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy of $F$. We first resolve a fundamental question of minimizing the number of cliques of size $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait, and Timmons. We also go further and prove a corresponding stability result. Next we minimize the number of cycles of length $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, and classify the extremal graphs for most values of $r$, answering another question of Kritschgau, Methuku, Tait, and Timmons for most $r$. We then move on to a central and longstanding conjecture in graph saturation made by Tuza, which states that for every graph $F$, the limit $\lim_{n \rightarrow \infty} \frac{\sat(n, F)}{n}$ exists, where $\sat(n, F)$ denotes the minimum number of edges in an $n$-vertex $F$-saturated graph. Pikhurko made progress in the negative direction by considering families of graphs instead of a single graph, and proved that there exists a graph family $\mathcal{F}$ of size $4$ for which $\lim_{n \rightarrow \infty} \frac{\sat(n, \mathcal{F})}{n}$ does not exist (for a family of graphs $\mathcal{F}$, a graph $G$ is called $\mathcal{F}$-saturated if $G$ does not contain a copy of any graph in $\mathcal{F}$, but the addition of any edge creates a copy of a graph in $\mathcal{F}$, and $\sat(n, \mathcal{F})$ is defined similarly). We make the first improvement in 15 years by showing that there exist infinitely many graph families of size $3$ where this limit does not exist. Our construction also extends to the generalized saturation problem when we minimize the number of fixed-size cliques.