Coloring complexes and arrangements

TL;DR

This paper links coloring and unipolar complexes to hyperplane arrangements, providing new proofs and restrictions on chromatic polynomials through convex ear decompositions, with implications for graph theory and matroid characteristic polynomials.

Contribution

It offers a novel interpretation of coloring and unipolar complexes via hyperplane arrangements, simplifying proofs and establishing new restrictions on chromatic polynomials.

Findings

All coloring complexes and many unipolar complexes have convex ear decompositions.

Convex ear decompositions impose new restrictions on chromatic polynomials of finite graphs.

Results extend to characteristic polynomials of submatroids of type B_n arrangements.

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.