论文标题
完整的两分图在飞机上灵活
Complete bipartite graphs flexible in the plane
论文作者
论文摘要
完整的两分图$ k_ {3,3} $，被视为平面链接，在顶点和杆作为边缘作为边缘，通常仅承认整个动作，即不灵活。迪克森（Dixon）在1899年发现了两种类型的悖论移动性。后来，在不同作者的一系列论文中，几乎所有对（m，n）$的灵活性问题$ k_ {m，n} $都解决了。在本文中，我们为欧几里得平面以及球体和双曲机平面中的所有完整两分图解决了它。我们提供独立的独立证明，而没有广泛的计算，这些计算在欧几里得，双曲线和球形案例中几乎相同。
A complete bipartite graph $K_{3,3}$, considered as a planar linkage with joints at the vertices and with rods as edges, in general admits only motions as a whole, i.e., is inflexible. Two types of its paradoxical mobility were found by Dixon in 1899. Later on, in a series of papers by different authors, the question of flexibility of $K_{m,n}$ was solved for almost all pairs $(m,n)$. In the present paper, we solve it for all complete bipartite graphs in the Euclidean plane as well as in the sphere and in the hyperbolic plane. We give independent self-contained proofs without extensive computations which are almost the same in the Euclidean, hyperbolic and spherical cases.
