Coloring complexes and arrangements

Abstract

Abstract. Steingrimsson's coloring complex and Jonsson's unipolar complex are interpreted in terms of hyperplane arrangements.This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type B n arrangements.

Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes

“…Jakob Jonsson proved [13] that ∆ G is homotopy equivalent to a wedge of spheres in fixed dimension, with the number of spheres being one less than the number of acyclic orientations of G. Axel Hultman [12] proved that ∆ G , and in general any link complex for a sub-arrangement of the type A or type B Coxeter arrangement, is shellable. Further, ∆ G admits a convex ear decomposition, as shown by Patricia Hersh and Ed Swartz [10].…”

Scheduling problems

“…It was shown that the generalized permutahedron nc-quasisymmetric function is L-positive and Γ(S) is h-positive. Returning to the graphical case, again in [15], the perspective of the normal fan is used to prove that the coloring complex has a convex ear decomposition which implies strong relations on the chromatic polynomial. The authors consider the generalization of their results to characteristic polynomials of matroids.…”

An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

“…For example, the complex in Figure 1, consisting of the boundary of the cube and the two hyperplanes, has a convex ear decomposition: Start with the boundary of the cube as triangulated by the braid arrangement, glue in the triangulated square lying on one of the hyperplanes and then glue in the two triangles on the second hyperplane one after the other. If all simplices in this complex are unimodular (as in many combinatorial applications), this leads to the following bounds, which have been successfully applied to the chromatic polynomial by Hersh and Swartz [38] and to the integral and modular flow and tension polynomials by Breuer and Dall [17].…”