Acyclic Orientations and the Chromatic Polynomial of Signed Graphs

We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial \(\chi_G(k,l)\) that counts the number of signed colorings using colors \(0,\pm1,\dots,\pm k\) along with \(l-1\) symmetric colors \(0_1,\dots,0_{l-1}\). We show that the evaluation of the bivariate chromatic polynomial \(|\chi_G(-1,2)|\) is equal to the number of acyclic orientations of the signed graph modulo the equivalence relation generated by swapping sources and sinks. We present three proofs of this fact, a proof using toric hyperplane arrangements, a proof using deletion-contraction, and a direct proof.