On the complexity of generalized chromatic polynomials

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 χ(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 CPcolorings 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.

Harary polynomials

“…(ii) Let H be connected of order t. DU (H) consist of non-empty disjoint unions of copies of H. DU (H)-colorings are mcc t -colorings. They are studied in [18]. (iii) Let F r(H) be class of H-free graphs.…”

“…(vi) If P consists of all connected graphs, we speak of convex colorings, and put C(G; x) = χ P (G; k), see [37,38,18].…”

“…Checking whether a graph G is k-colorable is NP-complete. For the complexity of checking whether a graph is P-colorable for various graph properties P, the reader may consult [10,1,18]. However, using Courcelle's celebrated Theorem, [14,Chapter 13] and [16,Chapter 11], MSOL-definability implies that checking whether a graph G is kcolorable is fixed parameter tractable (FPT) for graphs of bounded tree-width, and even for graphs of bounded clique-width or rank-width.…”

“…In future research we continue the study of the complexity of evaluating Harary polynomials, initiated in [18].…”