Overview
This research investigates a specific class of even $\Delta$-matroids, focusing on those derived from orientable ribbon graphs. The study introduces the concept of pseudo-orientable ribbon graphs to characterize strong $\Delta$-matroids that correspond to orientable ribbon-graphic $\Delta$-matroids. A geometric construction is presented to transform pseudo-orientable ribbon graphs into orientable ribbon graphs, establishing this correspondence. Consequences of this framework include the derivation of the Matrix--Quasi-tree Theorem, the Hurwitz stability of quasi-tree generating polynomials, and a log-concavity result related to quasi-trees in pseudo-orientable ribbon graphs. The log-concavity result extends Stanley's theorem for regular matroids to regular $\Delta$-matroids. The study also identifies an infinite family of ribbon graphs that are not pseudo-orientable and fail to satisfy the Matrix--Quasi-tree Theorem and Hurwitz stability.
Research Context
Even $\Delta$-matroids represent an important class within matroid theory, with orientable ribbon graphs serving an analogous role to graphic matroids. A natural correspondence exists between strong $\Delta$-matroids and even $\Delta$-matroids, as established by Geelen and Murota. This work is motivated by this correspondence, seeking to delineate the subset of strong $\Delta$-matroids that are specifically linked to orientable ribbon-graphic $\Delta$-matroids.
Approach
The research characterizes the relevant strong $\Delta$-matroids by associating them with what are termed pseudo-orientable ribbon graphs. To concretize this association, a geometric construction is provided. This construction explicitly transforms a pseudo-orientable ribbon graph into an orientable ribbon graph, thereby realizing the established correspondence between the two types of $\Delta$-matroids. The approach further involves extending Stanley's log-concavity theorem, originally applicable to regular matroids, to the broader context of regular $\Delta$-matroids. The study also investigates cases where the derived theorems do not hold, identifying an infinite family of non-pseudo-orientable ribbon graphs that fail to satisfy the Matrix--Quasi-tree theorem and Hurwitz stability.
Findings
- The class of strong $\Delta$-matroids corresponding to orientable ribbon-graphic $\Delta$-matroids are precisely those associated with pseudo-orientable ribbon graphs.
- A geometric construction exists that transforms a pseudo-orientable ribbon graph into an orientable ribbon graph, thereby realizing the correspondence between these graph types and their associated $\Delta$-matroids.
- The framework yields the Matrix--Quasi-tree Theorem.
- The framework yields the Hurwitz stability of quasi-tree generating polynomials.
- A log-concavity result is obtained for the sequence counting quasi-trees of size $2i-1$ or $2i$ for pseudo-orientable ribbon graphs.
- Stanley's log-concavity theorem for regular matroids can be generalized to regular $\Delta$-matroids.
- An infinite family of non-pseudo-orientable ribbon graphs has been identified that do not satisfy the Matrix--Quasi-tree theorem and Hurwitz stability.