Abstract In three influential papers in the 1980s and early 1990s, Joe Kung laid the foundations for extremal matroid theory which he envisaged as finding the growth rate of certain classes of matroids along with a characterisation of the extremal matroids in each such class. At the time, he was particularly interested in the minor-closed classes of binary matroids obtained by excluding the cycle matroids of the Kuratowski graphs K 3 , 3 and/or K 5 . While he obtained strong bounds on the growth rate of these classes, it seems difficult to give the exact growth rate without a complete characterisation of the matroids in each class, which at the time seemed hopelessly complicated. Many years later, Mayhew, Royle and Whittle gave a characterisation of the internally 4-connected binary matroids with no M ( K 3 , 3 ) -minor, from which the answers to Kung's questions follow immediately. In this characterisation, two thin families of binary matroids play an unexpectedly important role as the only non-cographic infinite families of internally 4-connected binary matroids with no M ( K 3 , 3 ) -minor. As the matroids are closely related to the cubic and quartic Mobius ladders, they were called the triangular Mobius matroids and the triadic Mobius matroids. Preliminary investigations of the class of binary matroids with no M ( K 5 ) -minor suggest that, once again, the triangular Mobius matroids will be the extremal internally 4-connected matroids in this class. Here we undertake a systematic study of these two families of binary matroids collecting in one place fundamental information about them, including their representations, connectivity properties, minor structure, automorphism groups and their chromatic polynomials. Along the way, we highlight the different ways in which these matroids have arisen naturally in a number of results and problems (both open and settled) in structural and extremal matroid theory.

中文翻译：

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莫比乌斯拟阵

摘要 在 1980 年代和 1990 年代初期的三篇有影响力的论文中，Joe Kung 奠定了极值拟阵理论的基础，他设想将其设想为找到某些类别的拟阵的增长率以及每个此类中的极值拟阵的特征。当时，他对通过排除 Kuratowski 图 K 3 、3 和/或 K 5 的圈拟阵而获得的二元拟阵的小闭类特别感兴趣。虽然他对这些类的增长率有很强的限制，但如果没有对每个类中拟阵的完整表征，似乎很难给出确切的增长率，这在当时似乎非常复杂。许多年后，Mayhew、Royle 和 Whittle 给出了没有 M ( K 3 , 3 ) -小调的内部 4 连通二元拟阵的表征，孔的问题的答案立即随之而来。在这个表征中，两个薄的二元拟阵家族扮演了一个出人意料的重要角色，作为唯一一个没有 M ( K 3 , 3 ) -次要的内部 4 连通二元拟阵的非共形无限家族。由于拟阵与三次方和四次方的莫比乌斯梯关系密切，故称为三角莫比乌斯拟阵和三方莫比乌斯拟阵。对没有 M ( K 5 ) -minor 的二元拟阵类的初步研究表明，三角形莫比乌斯拟阵将再次成为此类中的极值内部 4 连通拟阵。在这里，我们对这两个二元拟阵家族进行了系统的研究，在一个地方收集了关于它们的基本信息，包括它们的表示、连通性、次要结构、自同构群及其色多项式。在此过程中，我们强调了这些拟阵在结构和极值拟阵理论的许多结果和问题（开放和解决）中自然产生的不同方式。