A bivariate chromatic polynomial for signed graphs

We study Dohmen–P"onitz–Tittmann’s bivariate chromatic polynomial $c_{\mathrm{\Gamma }}(k,l)$ which counts all $(k+l)$-colorings of a graph $\mathrm{\Gamma }$ such that adjacent vertices get different colors if they are $\le k$. Our first contribution is an extension of $c_{\mathrm{\Gamma }}(k,l)$ to signed graphs, for which we obtain an inclusion–exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for $c_{\mathrm{\Gamma }}(k,l)$ and its signed-graph analogues, reminiscent of Stanley’s reciprocity theorem linking chromatic polynomials to acyclic orientations.