AI 中文总结
本文针对$d=3$的刚性拟阵秩函数开放问题，研究$K_t$-覆盖图，引入顶点秩贡献概念，证明相关猜想并得到$K_4$、$K_5$-覆盖图刚性的新连通性条件。
AI 中文摘要
在${\bf R}^d中一般刚性图$G$的刻画问题，或更一般地确定任意图$G$的$d$维刚性拟阵${\boldsymbol{\textit{R}}}_d(G)$的秩函数问题，当$d{\boldsymbol{\textit{≤}}}2$时已得到解决，但当$d{\boldsymbol{\textit{≥}}}3$时是离散几何领域的重大开放问题。本文聚焦$d=3$的情形，首先重新研究Dress在1987年提出的猜想：图$G$的${\boldsymbol{\textit{R}}}_3$闭包的秩由其所有大小至少为5的极大完全子图决定，我们证明该猜想给出的闭包秩值是实际值的上界，且若猜想成立将为所有图$G$的${\boldsymbol{\textit{R}}}_3(G)$秩提供良好刻画。Dress猜想中的秩公式引导我们研究$K_t$-覆盖图族，即每条边都属于某个大小$t{\boldsymbol{\textit{≥}}}3$的完全子图$K_t$的图，该族包含体-钉图、组合沸石、分子图等多个已被深入研究的图类。我们引入图$G$边集上任意拟阵中顶点的秩贡献这一新概念，并用其得到$K_t$-覆盖图$G$的${\boldsymbol{\textit{R}}}_3(G)$和${\boldsymbol{\textit{C}}}^1_2(G)$中顶点秩贡献的下界，利用这些下界证明体-钉图在${\boldsymbol{\textit{R}}}_3$中的秩的一个猜想的极小-极大公式对$C_2^1$-余因子拟阵成立（Whiteley猜想该拟阵等于${\boldsymbol{\textit{R}}}_3$），并得到${\boldsymbol{\textit{R}}}^3$中$K_4$-和$K_5$-覆盖图整体刚性的新的充分连通性条件。
英文摘要
The problems of characterizing the graphs $G$ which are generically rigid in ${\mathbb R}^d$, or more generally, determining the rank function of the $d$-dimensional rigidity matroid ${\cal R}_d(G)$ of an arbitrary graph $G$, have been solved when $d\leq 2$ but are major open problems in discrete geometry when $d\geq 3$. In this paper we shall concentrate on the case when $d=3$. We first revisit a conjecture of Dress from 1987 that the rank of the ${\cal R}_3$-closure of a graph $G$ is determined by its maximal complete subgraphs of size at least five. We show that his conjectured value for the rank of the closure gives an upper bound on the actual value. We also deduce that the truth of this conjecture would imply a good characterization of the rank of ${\cal R}_3(G)$ for all graphs $G$. The rank formula in Dress's conjecture leads us to consider the family of $K_t$-covered graphs, i.e., graphs in which every edge belongs to a complete subgraph $K_t$, for some $t\geq 3$. This family contains several well-studied graph classes such as body-pin graphs, combinatorial zeolites, and molecular graphs. We introduce a new notion of rank contributions of vertices in an arbitrary matroid on the edge set of a graph $G$, and use it to obtain lower bounds on the rank contributions of vertices in ${\cal R}_3(G)$ and ${\cal C}^1_2(G)$ when $G$ is $K_t$-covered. We use these bounds to show that a conjectured min-max formula for the rank of body-pin graphs in ${\cal R}_3$ holds for the $C_2^1$-cofactor matroid (which is conjectured by Whiteley to be equal to ${\cal R}_3$), and to obtain new sufficient connectivity conditions for the (global) rigidity of $K_4$- and $K_5$-covered graphs in ${\mathbb R}^3$.
