Algebras, Synchronous Games and Chromatic Numbers of Graphs

TL;DR

This paper introduces an algebraic framework for analyzing synchronous games and graph coloring problems, linking algebraic properties to quantum and classical chromatic numbers, and enabling computational approaches via Grobner bases.

Contribution

It develops a novel algebraic approach to characterize quantum and classical chromatic numbers of graphs through ideals and Grobner basis methods.

Findings

Algebraic characterizations of quantum chromatic numbers.

Connection between algebraic ideals and graph coloring properties.

Potential for computational analysis using Grobner basis techniques.

Abstract

We associate to each synchronous game an algebra whose representations determine if the game has a perfect deterministic strategy, perfect quantum strategy or one of several other perfect strategies. when applied to the graph coloring game, this leads to characterizations in terms of properties of an algebra of various quantum chromatic numbers that have been studied in the literature. This allows us to develop a correspondence between various chromatic numbers of a graph and ideals in this algebra which can then be approached via various Grobner basis methods.