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In the 1970's, Lov\'asz built a bridge between graphs and alternating matrixspaces, in the context of perfect matchings (FCT 1979). A similar connectionbetween bipartite graphs and matrix spaces plays a key role in the recentresolutions of the non-commutative rank problem(Garg-Gurvits-Oliveira-Wigderson, FOCS 2016; Ivanyos-Qiao-Subrahmanyam, ITCS2017). In this paper, we lay the foundation for another bridge between graphsand alternating matrix spaces, in the context of independent sets and vertexcolorings. The corresponding structures in alternating matrix spaces areisotropic spaces and isotropic decompositions, both useful structures in grouptheory and manifold theory.  We first show that the maximum independent set problem and the vertexc-coloring problem reduce to the maximum isotropic space problem and theisotropic c-decomposition problem, respectively. Next, we show that severaltopics and results about independent sets and vertex colorings have naturalcorrespondences for isotropic spaces and decompositions. These includealgorithmic problems, such as the maximum independent set problem for bipartitegraphs, and exact exponential-time algorithms for the chromatic number, as wellas mathematical questions, such as the number of maximal independent sets, andthe relation between the maximum degree and the chromatic number. Theseconnections lead to new interactions between graph theory and algebra. Someresults have concrete applications to group theory and manifold theory, and weinitiate a variant of these structures in the context of quantum informationtheory. Finally, we propose several open questions for further exploration.  This paper is dedicated to the memory of Ker-I Ko.

From independent sets and vertex colorings to isotropic spaces and isotropic decompositions