Vector Colorings of Random, Ramanujan, and Large-Girth Irregular Graphs

Jess Banks; Luca Trevisan

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arXiv · 1907.02539

Vector Colorings of Random, Ramanujan, and Large-Girth Irregular Graphs

Jess Banks·Luca Trevisan

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Abstract

We prove that in sparse Erd\H{o}s-R\'{e}nyi graphs of average degree $d$, the vector chromatic number (the relaxation of chromatic number coming from the Lov\`{a}sz theta function) is typically $\tfrac{1}{2}\sqrt{d} + o_d(1)$. This fits with a long-standing conjecture that various refutation and hypothesis-testing problems concerning $k$-colorings of sparse Erd\H{o}s-R\'{e}nyi graphs become computationally intractable below the `Kesten-Stigum threshold' $d_{KS,k} = (k-1)^2$. Along the way, we use the celebrated Ihara-Bass identity and a carefully constructed non-backtracking random walk to prove two deterministic results of independent interest: a lower bound on the vector chromatic number (and thus the chromatic number) using the spectrum of the non-backtracking walk matrix, and an upper bound dependent only on the girth and universal cover. Our upper bound may be equivalently viewed as a generalization of the Alon-Boppana theorem to irregular graphs

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Jess Banks, Luca Trevisan. 2019-07-04. Vector Colorings of Random, Ramanujan, and Large-Girth Irregular Graphs. https://arxiv.org/abs/1907.02539

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