The intersection numbers of moduli spaces of stable curves $\overline{\mathcal{M}}_{g,m}$ are well-studied and are known to have rich combinatorial structure. We introduce a natural class of these intersection numbers $ω_{G,g,m}$ indexed by finite simple graphs $G=(V,E)$. In genus zero, these numbers are closely related to several previously-studied quantities, including maximum likelihood degrees in algebraic statistics, counts of regions of certain hyperplane arrangements, and Kapranov degrees. We give two proofs of a simple closed formula $ω_{G,g,m}=(-1)^{\left\lvert V \right\rvert}χ_G(-(2g-2+m)),$ where $χ_G$ is the chromatic polynomial of $G$ -- one proof via intersection theory on moduli spaces of stable curves, and the other using the theory of hyperplane arrangements. We discuss several related questions and speculations, including new candidates for the chromatic polynomial of a directed graph.