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arXiv · 2610.22971

Sharp asymptotics for the Lovász number of random circulant graphs

Xin Chen·Hongwei Hu·Yanqing Hu·Wei Wang·Luzhen Yang
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Abstract

We study the LOVÁSZ number of dense random circulant graphs generated by independent Bernoulli bits indexed by cyclic distances. Although only $O(n)$ independent random inputs determine $O(n^2)$ globally dependent edge indicators, we prove sharp expectation asymptotics along all odd orders, including composite orders, with leading constant one. This resolves a conjecture posed for random circulant graphs in 2025. The proof combines an exact Fourier reduction and complementary duality with a positive interior smoothing that reveals a nonlinear random fixed-point equation. To analyze this equation under the strong dependence of the distance-bit model, we construct canonical tree fields, establish growing-degree Gaussian approximations through polynomial comparison and Wick/Hermite expansions, and transfer the resulting approximation back to the original sign model. A deterministic residual estimate together with an explicit exceptional-event integration yields the sharp expectation asymptotics.

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Xin Chen, Hongwei Hu, Yanqing Hu, Wei Wang, Luzhen Yang. 2026-10-16. Sharp asymptotics for the Lovász number of random circulant graphs. https://arxiv.org/abs/2610.22971

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