Coloring 3-Colorable Graphs with Low Threshold Rank
Jun-Ting Hsieh

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 -colorable graphs with small -sided threshold rank. Specifically, given an -vertex -colorable graph whose uniform random walk matrix has at most eigenvalues larger than , our algorithm finds a proper -coloring on at least vertices in time . This extends and improves upon the result of Bafna, Hsieh, and Kothari on -sided expanders. Furthermore, an independent work by Buhai, Hua, Steurer, and V\'ari-Kakas shows that it is UG-hard to properly -color more than vertices, thus establishing the tightness of our result. Our proof is short and simple, relying on the observation that for any distribution over proper -colorings, the correlation across an edge must be large if the marginals of the endpoints…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Complexity and Algorithms in Graphs · Markov Chains and Monte Carlo Methods
