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arXiv · 2507.20847

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

Chaithra P·Shushma Rani·R. Venkatesh
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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.

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Chaithra P, Shushma Rani, R. Venkatesh. 2025-07-28. Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements. https://arxiv.org/abs/2507.20847

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