Abstract
J. Makowsky and B.
Zilber (2004) showed that many variations of graph colorings, called CP-colorings in the sequel, give rise to graph polynomials. This is true in particular for harmonious colorings, convex colorings, mcc(t)-colorings, and rainbow colorings, and many more.
N. Linial (1986) showed that the chromatic polynomial chi(G; X) is #P-hard to evaluate for all but three values X = 0, 1, 2, where evaluation is in P.
This dichotomy includes evaluation at real or complex values, and has the further property that the set of points for which evaluation is in P is finite. We investigate how the complexity of evaluating univariate graph polynomials that arise from CP-colorings varies for different evaluation points.
We show that for some CP-colorings (harmonious, convex) the complexity of evaluation follows a similar pattern to the chromatic polynomial. However, in other cases (proper edge colorings, mcc(t)-colorings, H-free colorings) we could only obtain a dichotomy for evaluations at non-negative integer points.
We also discuss some CP-colorings where we only have very partial results.