A bivariate chromatic polynomial for signed graphs

We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial $c_\Gamma(k,l)$ which counts all $(k+l)$-colorings of a graph $\Gamma$ such that adjacent vertices get different colors if they are $\le k$. Our first contribution is an extension of $c_\Gamma(k,l)$ 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 $c_\Gamma(k,l)$ and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynomials to acyclic orientations.Comment: 8 pages, 4 figure

Boolean complexes for Ferrers graphs

In this paper we provide an explicit formula for calculating the boolean number of a Ferrers graph. By previous work of the last two authors, this determines the homotopy type of the boolean complex of the graph. Specializing to staircase shapes, we show that the boolean numbers of the associated Ferrers graphs are the Genocchi numbers of the second kind, and obtain a relation between the Legendre-Stirling numbers and the Genocchi numbers of the second kind. In another application, we compute the boolean number of a complete bipartite graph, corresponding to a rectangular Ferrers shape, which is expressed in terms of the Stirling numbers of the second kind. Finally, we analyze the complexity of calculating the boolean number of a Ferrers graph using these results and show that it is a significant improvement over calculating by edge recursion.Comment: final version, to appear in the The Australasian Journal of Combinatoric