Upper-Bounding the $k$-Colorability Threshold by Counting Covers
Let $G(n,m)$ be the random graph on $n$ vertices with $m$ edges. Let $d=2m/n$ be its average degree. We prove that $G(n,m)$ fails to be $k$-colorable with high probability if $d>2k\ln k-\ln k-1+o_k(1)$. This matches a conjecture put forward on the basis of sophisticated but non-rigorous statistical physics ideas (Krzakala, Pagnani, Weigt: Phys. Rev. E 70 (2004)). The proof is based on applying the first moment method to the number of "covers", a physics-inspired concept. By comparison, a standard first moment over the number of $k$-colorings shows that $G(n,m)$ is not $k$-colorable with high probability if $d>2k\ln k-k$.