Chain characteristic polynomials of matroids
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909726547116032 |
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| author | Lazzaro, Gary Wakefield, Max Weiss, Jason |
| author_facet | Lazzaro, Gary Wakefield, Max Weiss, Jason |
| contents | In this note we introduce a family of polynomials on a matroid derived from chain Tutte polynomials which generalize the classic and ubiquitous characteristic polynomial. We show that the coefficients of these polynomials alternate and present a recursion for any matroid. The classic characteristic polynomials were motivated by the chromatic polynomial on graphs. We define a generalized proper vertex coloring on a graph, which we call coupled multicoloring. We also define a generalized nowhere zero flow, which we call coupled multicommodity flow. Then we show that these chain characteristic polynomials enumerate the number of coupled multicolorings and coupled multicommodity flows on a graph. We conclude by listing multiple problems on enumerative properties of chain characteristic polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04789 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chain characteristic polynomials of matroids Lazzaro, Gary Wakefield, Max Weiss, Jason Combinatorics 05B35 (primary) 52B40, 05C15 (secondary) In this note we introduce a family of polynomials on a matroid derived from chain Tutte polynomials which generalize the classic and ubiquitous characteristic polynomial. We show that the coefficients of these polynomials alternate and present a recursion for any matroid. The classic characteristic polynomials were motivated by the chromatic polynomial on graphs. We define a generalized proper vertex coloring on a graph, which we call coupled multicoloring. We also define a generalized nowhere zero flow, which we call coupled multicommodity flow. Then we show that these chain characteristic polynomials enumerate the number of coupled multicolorings and coupled multicommodity flows on a graph. We conclude by listing multiple problems on enumerative properties of chain characteristic polynomials. |
| title | Chain characteristic polynomials of matroids |
| topic | Combinatorics 05B35 (primary) 52B40, 05C15 (secondary) |
| url | https://arxiv.org/abs/2508.04789 |