# Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs

arXiv: https://arxiv.org/abs/2008.12237 · HTML: https://mlsanity.com/paper/2008.12237/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Aug 2020.

## Built on by (most cited first, 13 of 13)

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- [Precise Error Rates for Computationally Efficient Testing](https://mlsanity.com/paper/2311.00289/) (2023)
- [Phase Transition for Stochastic Block Model with more than √n Communities (II)](https://mlsanity.com/paper/2511.21526/) (2025)
- [Low coordinate degree algorithms II: Categorical signals and generalized stochastic block models](https://mlsanity.com/paper/2412.21155/) (2024)
- [Phase Transition for Stochastic Block Model with more than √n Communities](https://mlsanity.com/paper/2509.15822/) (2025)
- [Low-degree phase transitions for detecting a planted clique in sublinear time](https://mlsanity.com/paper/2402.05451/) (2024)
- [Computational and statistical lower bounds for low-rank estimation under general inhomogeneous noise](https://mlsanity.com/paper/2510.08541/) (2025)

## Abstract (arXiv)

We study the problem of efficiently refuting the k-colorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, showing optimality of a simple spectral certificate. This evidence takes the form of a computationally-quiet planting: we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishable by a large class of algorithms. We generalize our results to the more general problem of certifying an upper bound on the maximum k-cut. This quiet planting is achieved by minimizing the effect of the planted structure (e.g. colorings or cuts) on the graph spectrum. Specifically, the planted structure corresponds exactly to eigenvectors of the adjacency matrix. This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. To illustrate this further, we give similar results for a Gaussian analogue of this problem: a quiet version of the spiked model, where we plant an eigenspace rather than adding a generic low-rank perturbation. Our evidence for computational hardness of distinguishing two distributions is based on three different heuristics: stability of belief propagation, the local statistics hierarchy, and the low-degree likelihood ratio. Of independent interest, our results include general-purpose bounds on the low-degree likelihood ratio for multi-spiked matrix models, and an improved low-degree analysis of the stochastic block model.