Robust Factorizations and Colorings of Tensor Graphs

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 883-916, March 2024.
Abstract. Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around rounding the solution to a Semidefinite Program. However, it is likely that important combinatorial or algebraic insights are needed in order to break the [math] threshold. One way to develop new understanding in graph coloring is to study special subclasses of graphs. For instance, Blum studied the 3-coloring of random graphs, and Arora and Ge studied the 3-coloring of graphs with low threshold-rank. In this work, we study graphs that arise from a tensor product, which appear to be novel instances of the 3-coloring problem. We consider graphs of the form [math] with [math] and [math], where [math] is any edge set such that no vertex has more than an [math]-fraction of its edges in [math]. We show that one can construct [math] with [math] that is close to [math]. For arbitrary [math], [math] satisfies [math]. Additionally, when [math] is a mild expander, we provide a 3-coloring for [math] in polynomial time. These results partially generalize an exact tensor factorization algorithm of Imrich. On the other hand, without any assumptions on [math], we show that it is NP-hard to 3-color [math].