Maximum star densities

TL;DR

This paper extends the understanding of maximum star densities in graphs with a given edge density, showing that extremal configurations are similar to cliques or their complements for all star sizes.

Contribution

It generalizes previous results for $k=2$ to all integers $k \u2265 2$, identifying extremal graph structures for maximum $k$-edge star counts at given densities.

Findings

Maximum star counts are asymptotically achieved by clique or complement structures.

The result applies uniformly for all star sizes $k \u2265 2$.

Extends prior work from $k=2$ to all $k \u2265 2$.

Abstract

Given an integer $k \geq 2$ and a real number $\gamma\in [0, 1]$, which graphs of edge density $\gamma$ contain the largest number of $k$-edge stars? For $k=2$ Ahlswede and Katona proved that asymptotically there cannot be more such stars than in a clique or in the complement of a clique (depending on the value of $\gamma$). Here we extend their result to all integers $k\ge 2$.