Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements
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| Format: | Preprint |
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2025
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| _version_ | 1866909709209960448 |
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| author | P, Chaithra Rani, Shushma Venkatesh, R. |
| author_facet | P, Chaithra Rani, Shushma Venkatesh, R. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_20847 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements P, Chaithra Rani, Shushma Venkatesh, R. Combinatorics Number Theory 05C15, 52C35 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. |
| title | Marked multi-colorings and marked chromatic polynomials of hypergraphs and subspace arrangements |
| topic | Combinatorics Number Theory 05C15, 52C35 |
| url | https://arxiv.org/abs/2507.20847 |