Coloring 3-Colorable Graphs with Low Threshold Rank

TL;DR

This paper introduces an efficient algorithm for coloring 3-colorable graphs with low threshold rank, nearly matching the theoretical hardness bounds, and simplifies the proof using correlation properties of colorings.

Contribution

It provides a new algorithm for large independent sets in 3-colorable graphs with small threshold rank, extending previous work and establishing tight bounds.

Findings

Algorithm finds proper 3-coloring on at least half the vertices.

Runs in time polynomial in graph size with dependence on threshold rank.

Proves the tightness of the coloring bound via hardness results.

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.