Boolean graphs are unmixed and vertex decomposable

TL;DR

This paper proves that Boolean graphs and their complements are vertex decomposable and unmixed, establishing their Cohen-Macaulay property, which advances understanding of their algebraic and combinatorial structure.

Contribution

It is shown that Boolean graphs and their complements are vertex decomposable and unmixed, a novel result linking graph properties with algebraic Cohen-Macaulayness.

Findings

Boolean graphs are vertex decomposable

Their complements are also vertex decomposable

Boolean graphs are unmixed and Cohen-Macaulay

Abstract

For each Boolean graph $B_n$, it is proved that both $B_n$ and its complement graph $\overline{B_n}$ are vertex decomposable. It is also proved that $B_n$ is an unmixed graph, thus it is also Cohen-Macaulay.