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arXiv · 2110.08950

A Sum of Squares Characterization of Perfect Graphs

Amir Ali Ahmadi·Cemil Dibek
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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.

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Amir Ali Ahmadi, Cemil Dibek. 2021-10-17. A Sum of Squares Characterization of Perfect Graphs. https://arxiv.org/abs/2110.08950

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