Graph colouring and Steenrod's problem for Stanley-Reisner rings

TL;DR

This paper explores the algebraic analogues of graph coloring and their relation to Steenrod's problem, focusing on Stanley-Reisner rings and their cohomology ring realizations.

Contribution

It introduces algebraic versions of graph coloring and connects span coloring to Steenrod's problem for Stanley-Reisner rings, providing new insights into their structure.

Findings

Defined algebraic graph coloring and span coloring.

Established relations between span coloring and Steenrod's problem.

Linked Stanley-Reisner rings to cohomology ring realizations.

Abstract

It is a classical problem in algebraic topology asked by Steenrod which graded rings occur as the cohomology ring of a space. In this paper, we define an algebraic version of the graph colouring, span colouring, and observe the relation between span colourings and Steenrod's problem for graded Stanley-Reisner rings, in other words polynomial rings divided by an ideal generated by square-free monic monomials.