SIAM Journal on Computing, Volume 50, Issue 3, Page 924-971, January 2021.
In the 1970s, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings [Proceedings of FCT, 1979, pp. 565--574]. A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the noncommutative rank problem [A. Garg et al., Proceedings of FOCS, 2016, pp. 109--117; G. Ivanyos, Y. Qiao, and K. V. Subrahmanyam, Comput. Complexity, 26 (2017), pp. 717--763]. In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex $c$-coloring problem reduce to the maximum isotropic space problem and the isotropic $c$-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.

中文翻译：

---

从独立集和顶点着色到各向同性空间和各向同性分解：图和交替矩阵空间之间的另一座桥梁

SIAM Journal on Computing，第 50 卷，第 3 期，第 924-971 页，2021 年 1 月。
在 1970 年代，Lovász 在完美匹配的背景下在图和交替矩阵空间之间架起了一座桥梁 [Proceedings of FCT, 1979, pp. 565--574]。二部图和矩阵空间之间的类似联系在非交换秩问题的最近解决中起着关键作用 [A. Garg 等人，《FOCS 会议录》，2016 年，第 109--117 页；G. Ivanyos、Y. Qiao 和 KV Subrahmanyam，Comput。复杂性，26 (2017)，第 717--763 页]。在本文中，我们在独立集和顶点着色的背景下，为图和交替矩阵空间之间的另一座桥梁奠定了基础。交替矩阵空间中的相应结构是各向同性空间和各向同性分解，它们都是群论和流形理论中的有用结构。我们首先证明最大独立集问题和顶点$c$-着色问题分别归结为最大各向同性空间问题和各向同性$c$-分解问题。接下来，我们展示了关于独立集和顶点着色的几个主题和结果对于各向同性空间和分解具有自然的对应关系。这些包括算法问题，例如二部图的最大独立集问题，色数的精确指数时间算法，以及数学问题，例如最大独立集的数量，以及最大度数和最大度数之间的关系。色数。这些联系导致了图论和代数之间的新相互作用。一些结果对群论和流形理论有具体的应用，我们在量子信息理论的背景下启动了这些结构的变体。最后，我们提出了几个开放性问题以供进一步探索。