A Sum of Squares Characterization of Perfect Graphs
Optimization and Control
2023-05-03 v5 Algebraic Geometry
Combinatorics
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.
Keywords
Cite
@article{arxiv.2110.08950,
title = {A Sum of Squares Characterization of Perfect Graphs},
author = {Amir Ali Ahmadi and Cemil Dibek},
journal= {arXiv preprint arXiv:2110.08950},
year = {2023}
}