Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements

TL;DR

This paper introduces marked multi-colorings and marked chromatic polynomials for hypergraphs and subspace arrangements, generalizing graph results and establishing polynomial properties and conjectures about their coefficients.

Contribution

It extends the concepts of chromatic polynomials and independence series to hypergraphs and subspace arrangements, providing new polynomial invariants and conjectures.

Findings

Coefficients of the q-th power of the marked independence series match the marked chromatic polynomials.

Number of marked multi q-colorings of a subspace arrangement is polynomial in q.

(-q)-th power of the independence series has non-negative coefficients for hyperplane arrangements.

Abstract

We introduce the concepts of marked multi-colorings, marked chromatic polynomials, and marked (multivariate) independence series for hypergraphs. We show that the coefficients of the q-th power of the marked independence series of a hypergraph coincide with its marked chromatic polynomials in q, thereby generalizing a corresponding result for graphs established in Chaithra et al. 2025 (arXiv:2503.11230). These notions are then naturally extended to subspace arrangements. In particular, we prove that the number of marked multi q-colorings of a subspace arrangement is a polynomial in q. We also define the (marked) independence series for subspace arrangements and prove that the (-q)-th power of the independence series of a hyperplane arrangement has non-negative coefficients. We further conjecture that the (-q)-th power of the independence series of a hypergraph has non-negative coefficients if and only if all its edges have even cardinality.