Pseudo-orientable ribbon graphs: Matrix--Quasi-tree Theorem and log-concavity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910042918223872 |
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| author | Ding, Changxin Kim, Donggyu |
| author_facet | Ding, Changxin Kim, Donggyu |
| contents | One of the most important classes of even $Δ$-matroids arises from orientable ribbon graphs, which play a role analogous to that of graphic matroids in matroid theory. Motivated by a natural correspondence between strong $Δ$-matroids and even $Δ$-matroids due to Geelen and Murota, we characterize the class of strong $Δ$-matroids that correspond to orientable ribbon-graphic $Δ$-matroids. These are precisely the $Δ$-matroids associated with what we call pseudo-orientable ribbon graphs. Moreover, we present a geometric construction that transforms a pseudo-orientable ribbon graph into an orientable ribbon graph, thereby realizing this correspondence.
As consequences, we obtain the Matrix--Quasi-tree Theorem, the Hurwitz stability of quasi-tree generating polynomials, and a log-concavity result for the sequence counting quasi-trees of size $2i-1$ or $2i$ for pseudo-orientable ribbon graphs. To establish the log-concavity, we generalize Stanley's log-concavity theorem for regular matroids to regular $Δ$-matroids. Finally, we exhibit an infinite family of non-pseudo-orientable ribbon graphs that fail to satisfy the Matrix--Quasi-tree theorem and Hurwitz stability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_05702 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pseudo-orientable ribbon graphs: Matrix--Quasi-tree Theorem and log-concavity Ding, Changxin Kim, Donggyu Combinatorics 05C10, 05C50, 05C30 One of the most important classes of even $Δ$-matroids arises from orientable ribbon graphs, which play a role analogous to that of graphic matroids in matroid theory. Motivated by a natural correspondence between strong $Δ$-matroids and even $Δ$-matroids due to Geelen and Murota, we characterize the class of strong $Δ$-matroids that correspond to orientable ribbon-graphic $Δ$-matroids. These are precisely the $Δ$-matroids associated with what we call pseudo-orientable ribbon graphs. Moreover, we present a geometric construction that transforms a pseudo-orientable ribbon graph into an orientable ribbon graph, thereby realizing this correspondence. As consequences, we obtain the Matrix--Quasi-tree Theorem, the Hurwitz stability of quasi-tree generating polynomials, and a log-concavity result for the sequence counting quasi-trees of size $2i-1$ or $2i$ for pseudo-orientable ribbon graphs. To establish the log-concavity, we generalize Stanley's log-concavity theorem for regular matroids to regular $Δ$-matroids. Finally, we exhibit an infinite family of non-pseudo-orientable ribbon graphs that fail to satisfy the Matrix--Quasi-tree theorem and Hurwitz stability. |
| title | Pseudo-orientable ribbon graphs: Matrix--Quasi-tree Theorem and log-concavity |
| topic | Combinatorics 05C10, 05C50, 05C30 |
| url | https://arxiv.org/abs/2603.05702 |