On rigidity of unit-bar frameworks

TL;DR

This paper demonstrates the existence of infinitesimally rigid bipartite unit-bar frameworks in any dimension and constructs planar frameworks with girth up to 12 that are also infinitesimally rigid, addressing open problems.

Contribution

It establishes the existence of infinitesimally rigid bipartite frameworks in arbitrary dimensions and constructs planar frameworks with high girth that are infinitesimally rigid, solving previously posed problems.

Findings

Existence of infinitesimally rigid bipartite frameworks in all dimensions.

Construction of planar frameworks with girth up to 12 that are infinitesimally rigid.

Addresses and resolves open problems by Maehara.

Abstract

We show the existence of infinitesimally rigid bipartite unit-bar frameworks in $\mathbb{R}^d$. We also construct unit-bar frameworks with girth up to 12 that are infinitesimally rigid in the plane. This answers problems proposed by Maehara.