Generalized saturation problems for cliques, paths, and stars

TL;DR

This paper investigates the minimum number of specific subgraphs in saturated graphs avoiding certain subgraphs, extending previous work to include cliques, stars, paths, and trees, with new bounds and relationships.

Contribution

It introduces generalized saturation parameters for various subgraphs, focusing on cliques, stars, paths, and trees, expanding the understanding of saturated graph structures.

Findings

Derived bounds for sat_{K_r}(n,S_t) and sat_{S_r}(n,S_t)

Extended results to paths and trees in saturated graphs

Connected to recent research on saturation numbers for complete graphs and stars

Abstract

A graph $G$ is $F$-saturated if it does not contain any copy of $F$, but the addition of any missing edge in $G$ creates at least one copy of $F$. Inspired by work of Alon and Shikhelman regarding a similar question for $F$-free graphs, Kritschgau, Methuku, Tait, and Timmons introduced the parameter of $\text{sat}_H(n,F)$ to denote the minimum number of copies of some subgraph $H$ in an $F$-saturated graph on $n$ vertices. In this paper, we address this generalized saturation problem with special focus on $\text{sat}_{K_r}(n,S_t)$ and $\text{sat}_{S_r}(n,S_t)$ This relates to recent work by Chakraborti and Loh regarding $\text{sat}_{K_r}(n,K_t)$ and by Ergemlidze, Methuku, Tait, and Timmons regarding $\text{sat}_{S_r}(n,K_t)$. We also provide some results regarding paths and arbitrary trees.