On the dimension of the minimal vertex covers semigroup ring of an unmixed bipartite graph

TL;DR

This paper investigates the relationship between the dimension of the semigroup ring of minimal vertex covers and the Boolean lattice rank in unmixed bipartite graphs, extending previous algebraic graph theory work.

Contribution

It establishes a connection between the semigroup ring dimension and the Boolean lattice rank for a class of bipartite graphs, providing new insights into their algebraic structure.

Findings

Dimension of the semigroup ring equals the Boolean lattice rank for unmixed bipartite graphs.

Provides a formula linking graph properties to algebraic invariants.

Extends previous work by Herzog, Hibi, and Ohsugi from 2008.

Abstract

In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the semigroup ring associated to the set of minimal vertex covers of an unmixed bipartite graph. In this paper we relate the dimension of this semigroup ring to the rank of the Boolean lattice associated to the graph.