A Sum of Squares Characterization of Perfect Graphs

TL;DR

This paper provides an algebraic sum of squares characterization of perfect graphs, linking graph properties to polynomial nonnegativity and sums of squares representations, offering new insights into graph theory and polynomial algebra.

Contribution

It introduces a novel algebraic criterion for perfect graphs using sums of squares polynomials, connecting graph theory with algebraic and polynomial methods.

Findings

Characterization of perfect graphs via sum of squares polynomials

Identification of nonnegative polynomials not representable as sums of squares

Reformulation of classical perfect graph results in algebraic terms

Abstract

We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative polynomials associated with the graph are sums of squares. As a byproduct, we obtain several infinite families of nonnegative polynomials that are not sums of squares through graph-theoretic constructions. We also characterize graphs for which the associated polynomials belong to certain structured subsets of sum of squares polynomials. Finally, we reformulate some well-known results from the theory of perfect graphs as statements about sum of squares proofs of nonnegativity of certain polynomials.