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arXiv · 2609.33684

Improved SDP Coloring of 3-Colorable Graphs from Recursive Gaussian Certificates

Ijay Narang·Yukai Tang
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Abstract

We give a randomized polynomial-time algorithm that, for every fixed $\varepsilon > 0$, colors every $3$-colorable $n$-vertex graph using $O\bigl(n^{(13-\sqrt{97})/18+\varepsilon}\bigr) \approx O\bigl(n^{0.17506+\varepsilon}\bigr)$ colors, improving upon the previous best bound of $O(n^{0.19539})$ from Bansal, Huang, and Lee. Our improvement comes from analyzing higher-level neighborhoods through a recursive description of failure in Gaussian SDP rounding. If the rounding returns too small an independent set, it produces local Gaussian certificates at every vertex of a nonempty induced subgraph. We propagate these certificates along walks to higher-level neighborhoods by defining a recursive certificate structure and proving a strengthened cover-composition lemma, which refines the one of Arora, Chlamt{á}{č}, and Charikar. We then construct a bounded potential function that increases by a fixed positive amount at every propagation step, yielding a contradiction. Consequently, the rounding must produce a sufficiently large independent set.

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Ijay Narang, Yukai Tang. 2026-09-27. Improved SDP Coloring of 3-Colorable Graphs from Recursive Gaussian Certificates. https://arxiv.org/abs/2609.33684

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