On the complexity of generalized chromatic polynomials

TL;DR

This paper studies the computational complexity of evaluating generalized graph coloring polynomials at various points, revealing patterns and differences across multiple coloring variants.

Contribution

It extends the complexity analysis of chromatic polynomials to a broad class of CP-colorings, identifying where evaluation is hard or tractable.

Findings

Harmonious and convex colorings have complexity patterns similar to the chromatic polynomial.

For some colorings, a full dichotomy of evaluation complexity is established at non-negative integers.

Partial results are obtained for certain other CP-colorings.

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.