The odd–even invariant for graphs

The odd–even invariant for graphs is the graphic version of the odd–even invariant for oriented matroids. Here, simple properties of this invariant are verified, and for certain graphs, including chordal graphs and complete bipartite graphs, its value is determined. The odd–even chromatic polynomial is introduced, its coefficients are briefly studied, and it is shown that the absolute value of this polynomial at −1 equals the odd–even invariant, in analogy with the usual chromatic polynomial and the number of acyclic orientations.