The Templates for Some Classes of Quaternary Matroids

Decomposition of Quaternary Signed-Graphic Matroids

In this work we provide a decomposition theorem for the class of quaternary and non-binary signed-graphic matroids. This generalizes previous results for binary signed-graphic matroids and graphic matroids, and it provides the theoretical basis for a recognition algorithm.

Quaternary matroids are vf-safe

Binary delta-matroids are closed under vertex flips, which consist of the natural operations of twist and loop complementation. In this note we provide an extension of this result from GF(2) to GF(4). As a consequence, quaternary matroids are "safe" under vertex flips (vf-safe for short). As an application, we find that the matroid of a bicycle space of a quaternary matroid is independent of the chosen representation. This extends a result of Vertigan [J. Comb. Theory B (1998)] concerning the bicycle dimension of quaternary matroids.

Quaternary Bicycle Matroids and the Penrose Polynomial for Delta-Matroids

In contrast to matroids, vf-safe delta-matroids have three kinds of minors and are closed under the operations of twist and loop complementation. We show that the delta-matroids representable over GF(4) with respect to the nontrivial automorphism of GF(4) form a subclass of the vf-safe delta-matroids closed under twist and loop complementation. In particular, quaternary matroids are vf-safe. Using this result, we show that the matroid of a bicycle space of a quaternary matroid M is obtained from M by using loop complementation. As a consequence, the matroid of a bicycle space of a quaternary m…

三元与四元正拟阵（positroids）

Ternary and quaternary positroids

该研究给出三元与四元正拟阵的禁用子式刻画，明确三元正拟阵的构造方式与四元正拟阵的禁用子式，并完整刻画三元正拟阵的格拉斯曼项链纤维结构。

四熵拟阵是四元的

Four-Entropic Matroids Are Quaternary

该研究证明4-熵拟阵等价于F₄可表示拟阵，结合相关结构性质完成证明，并将结论应用于理想完美访问结构的线性方案构造。

Templates for Binary Matroids

A binary frame template is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to conform or coconform to the template. We introduce a preorder on these templates and determine the nontrivial templates that are minimal with respect to this order. As an application of our main result, we determine the eventual growth rates of certain minor-closed classes of binary matroids, including the class of binary matroids with no minor isomorphic to PG(3,2). Our main result applies to all highly-connected matroids in a class, not just those of maximum size. As…