Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around SDP-based rounding. However, it is likely that important combinatorial or algebraic insights are needed in order to break the $n^{o(1)}$ 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 which arise from a tensor product, which appear to be novel instances of the 3-coloring problem. We consider graphs of the form $H = (V,E)$ with $V =V( K_3 \times G)$ and $E = E(K_3 \times G) \setminus E'$, where $E' \subseteq E(K_3 \times G)$ is any edge set such that no vertex has more than an $\epsilon$ fraction of its edges in $E'$. We show that one can construct $\widetilde{H} = K_3 \times \widetilde{G}$ with $V(\widetilde{H}) = V(H)$ that is close to $H$. For arbitrary $G$, $\widetilde{H}$ satisfies $|E(H) \Delta E(\widetilde{H})| \leq O(\epsilon|E(H)|)$. Additionally when $G$ is a mild expander, we provide a 3-coloring for $H$ in polynomial time. These results partially generalize an exact tensor factorization algorithm of Imrich. On the other hand, without any assumptions on $G$, we show that it is NP-hard to 3-color $H$.

翻译：自Karger、Motwani和Sudan的开创性结果以来，近似三着色算法主要围绕SDP舍入展开。然而，要突破$n^{o(1)}$阈值，很可能需要关键的组合或代数洞见。研究图的特殊子类是发展图着色新理解的一种途径。例如，Blum研究了随机图的三着色，Arora和Ge研究了低阈值秩图的三着色。本文研究由张量积产生的图，这似乎是三着色问题的新实例。我们考虑形如$H = (V,E)$的图，其中$V = V(K_3 \times G)$，$E = E(K_3 \times G) \setminus E'$，且$E' \subseteq E(K_3 \times G)$是任意边集，使得每个顶点在$E'$中的边数不超过其总边数的$\epsilon$比例。我们证明可以构造$\widetilde{H} = K_3 \times \widetilde{G}$，其中$V(\widetilde{H}) = V(H)$，且$\widetilde{H}$接近于$H$。对于任意$G$，$\widetilde{H}$满足$|E(H) \Delta E(\widetilde{H})| \leq O(\epsilon|E(H)|)$。此外，当$G$是温和扩张图时，我们能在多项式时间内给出$H$的三着色。这些结果部分推广了Imrich的精确张量分解算法。另一方面，在不假设$G$任何条件的情况下，我们证明对$H$进行三着色是NP难的。