Realization of some Stanley-Reisner algebras and graph colorings

TL;DR

This paper explores the realization of Stanley-Reisner algebras derived from graphs and connects their algebraic realizability to the graph's span coloring, advancing understanding in algebraic topology and combinatorics.

Contribution

It introduces graph-dependent Stanley-Reisner algebras and links their realizability to graph coloring, providing new insights into algebraic topology.

Findings

Stanley-Reisner algebras depend on graph structure

Realizability linked to span coloring of graphs

Provides criteria for algebraic realization based on graph coloring

Abstract

It is a classical problem in algebraic topology to decide whether a given graded $\mathbb{Z}$-algebra can be realized as the cohomology ring of a space. In this paper, we introduce families of Stanley-Reisner algebras depending on graphs, and relate their realizability to the span coloring of the graph.