Pseudo-orientable ribbon graphs: Matrix--Quasi-tree Theorem and log-concavity

Abstract

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.

Figures (14)

• Figure 1: A pseudo-orientable ribbon graph $\mathbb{G}$ and its adjustment $\widehat{\mathbb{G}}$, together with the corresponding matrices and $\Delta$-matroids. $\widehat{\mathbb{G}}$ is obtained by flipping the upper half-circle of the vertex of $\mathbb{G}$ and adding a new orientable loop, denoted by $4$, which connects the two boundary points of the intersection of the half-circles. The matrix $\mathbf{A}$ encodes the interlacements along the loops in $\widehat{\mathbb{G}} \setminus 4$, while the vector $\mathbf{v}$ records the non-orientable loops in the original $\mathbb{G}$. The associated $\Delta$-matroids $D$ and $\widehat{D}$ encode the quasi-trees of $\mathbb{G}$ and $\widehat{\mathbb{G}}$, respectively.
• Figure 2: A bouquet (left) with three loops and the corresponding signed chord diagram (right). The edges and chords colored blue indicate orientable loops (assigned $0$), and the ones colored red indicate non-orientable loops (assigned $1$).
• Figure 3: The partial dual of a bouquet $\mathbb{B}$ at a non-orientable loop $e$. The blue chords represent orientable loops, and the red chords represent non-orientable loops.
• Figure 4: The partial dual of a bouquet $\mathbb{B}$ at an interlacing pair of orientable loops $f$ and $f'$. The blue chords represent orientable loops, and the red chords represent non-orientable loops.
• Figure 5: The left figure is a pseudo-orientable bouquet $\mathbb{B}$ whose orientable loops are colored in blue and non-orientable loops are colored in red. A certificate $(S_1,S_2)$ is indicated by dashed arrows. The right figure is the adjustment $\widehat{\mathbb{B}}$ of $\mathbb{B}$ at $(S_1,S_2)$, which is obtained by flipping the bottom segment $S_2$. The new loop $\widehat{e}$ is depicted as the blue-green dashed line. All loops in $\widehat{\mathbb{B}}$ are orientable.

Theorems & Definitions (110)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.3
• Definition 2.4
• Definition 2.5: Partial duality
• Proposition 2.6: Chmutov2009
• Proposition 2.7: Chmutov2009