Coloring 3-Colorable Graphs with Low Threshold Rank

Abstract

We present a new algorithm for finding large independent sets in $3$-colorable graphs with small $1$-sided threshold rank. Specifically, given an $n$-vertex $3$-colorable graph whose uniform random walk matrix has at most $r$ eigenvalues larger than $\varepsilon$, our algorithm finds a proper $3$-coloring on at least $(\frac{1}{2}-O(\varepsilon))n$ vertices in time $n^{O(r/\varepsilon^2)}$. This extends and improves upon the result of Bafna, Hsieh, and Kothari on $1$-sided expanders. Furthermore, an independent work by Buhai, Hua, Steurer, and V\'ari-Kakas shows that it is UG-hard to properly $3$-color more than $(\frac{1}{2}+\varepsilon)n$ vertices, thus establishing the tightness of our result. Our proof is short and simple, relying on the observation that for any distribution over proper $3$-colorings, the correlation across an edge must be large if the marginals of the endpoints are not concentrated on any single color. Notably, this property fails for $4$-colorings, which is consistent with the hardness result of [BHK25] for $4$-colorable $1$-sided expanders.