Vector Colorings of Random, Ramanujan, and Large Girth Irregular Graphs
We prove that in sparse Erdos Renyi graphs of average degree $d$ the vector chromatic number (the relaxation of chromatic number coming from the Lovasz 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 Erdos Renyi 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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