Tight concentration of star saturation number in random graphs

TL;DR

This paper proves that in random graphs, the star saturation number is sharply concentrated around two consecutive values, refining previous asymptotic results for the minimum edges in maximal star-free subgraphs.

Contribution

The authors establish a precise concentration result for the star saturation number in random graphs, improving upon known asymptotic estimates.

Findings

Star saturation number in G(n,p) is concentrated in two consecutive points.

Provides a sharper probabilistic understanding of star-free subgraphs.

Refines previous asymptotic results for star saturation in random graphs.

Abstract

For given graphs $F$ and $G$, the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$ is called the $F$-saturation number and denoted $\mathrm{sat}(G, F)$. For the star $F=K_{1,r}$, the asymptotics of $\mathrm{sat}(G(n,p),F)$ is known. We prove a sharper result: whp $\mathrm{sat}(G(n,p), K_{1,r})$ is concentrated in a set of 2 consecutive points.