论文标题
由平行四边形组成的支撑框架
Bracing frameworks consisting of parallelograms
论文作者
论文摘要
平面中的矩形可以连续变形，以保留其边缘长度，但添加对角线支撑可以防止这种变形。 Bolker和Crapo以组合表征了牙套的选择，即使用支撑图制成无限型刚性的平方网格：一个两部分图，其顶点是网格的列和行，而行和列仅在且仅在支撑的正方形时就相邻。杜阿尔特（Duarte）和弗朗西斯（Francis）将支撑图的概念概括为菱形地毯，证明了支撑图的连通性意味着刚性，并在没有证据的情况下说明了另一个含义。纳吉·凯姆（Nagy Kem）在无限设置中给出了等效性。我们考虑由来自更通用类别的图组成的支撑框架的连续变形及其在平面中的位置，以使每个4周期形成平行四边形。我们表明，这种支撑框架的刚度等于特殊边缘着色的不存在，这反过来又等同于所连接的相应支撑图。
A rectangle in the plane can be continuously deformed preserving its edge lengths, but adding a diagonal brace prevents such a deformation. Bolker and Crapo characterized combinatorially which choices of braces make a grid of squares infinitesimally rigid using a bracing graph: a bipartite graph whose vertices are the columns and rows of the grid, and a row and column are adjacent if and only if they meet at a braced square. Duarte and Francis generalized the notion of the bracing graph to rhombic carpets, proved that the connectivity of the bracing graph implies rigidity and stated the other implication without proof. Nagy Kem gives the equivalence in the infinitesimal setting. We consider continuous deformations of braced frameworks consisting of a graph from a more general class and its placement in the plane such that every 4-cycle forms a parallelogram. We show that rigidity of such a braced framework is equivalent to the non-existence of a special edge coloring, which is in turn equivalent to the corresponding bracing graph being connected.
