论文标题
不是顶点传输的边缘传输多型
The Edge-transitive Polytopes that are not Vertex-transitive
论文作者
论文摘要
在三维欧几里得空间中，存在两个特殊的多面体，菱形十二面体和菱形三角体，这是唯一已知的多型（除了多边形），它们是边缘传输而不是顶点转换的。我们表明，这些多面体没有较高的类似物，也就是说，在尺寸$ d \ ge 4 $中，凸多型的边缘传递意味着顶点传播性。更一般而言，我们给出了所有凸多图形的分类，同时具有相同长度的所有边缘，边缘在球体内和两部分边缘图。我们表明，尺寸中的任何此类多层$ d \ ge 4 $都是顶点传递。
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, the only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show that these polyhedra do not have higher-dimensional analogues, that is, that in dimension $d\ge 4$, edge-transitivity of convex polytopes implies vertex-transitivity. More generally, we give a classification of all convex polytopes which at the same time have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show that any such polytope in dimension $d\ge 4$ is vertex-transitive.
