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

TL;DR

This paper establishes the typical vector chromatic number of sparse Erdős-Rényi graphs, relates it to computational hardness thresholds, and introduces new spectral bounds using non-backtracking walks and the Ihara-Bass identity.

Contribution

It provides the first precise asymptotic for the vector chromatic number in sparse Erdős-Rényi graphs and introduces novel spectral bounds based on non-backtracking walks and girth.

Findings

Vector chromatic number is approximately half the square root of average degree.

Spectral bounds relate vector chromatic number to non-backtracking walk spectrum.

Upper bounds generalize the Alon-Boppana theorem to irregular graphs.

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