5-regular graphs and the 3-dimensional rigidity matroid

TL;DR

This paper proves a conjecture that 5-regular graphs with certain edge constraints are independent in the 3D rigidity matroid, advancing understanding of rigidity in higher dimensions.

Contribution

It establishes that all 5-regular graphs satisfying specific subgraph edge conditions are independent in the 3D rigidity matroid, confirming a conjecture by Jackson and Jordán.

Findings

Proves the Jackson-Jordán conjecture for 5-regular graphs in 3D.

Characterizes independence in the 3D rigidity matroid for a new class of graphs.

Enhances understanding of combinatorial rigidity in higher dimensions.

Abstract

A bar-joint framework $(G,p)$ in Euclidean $d$-space is rigid if the only edge-length-preserving continuous motions arise from isometries of $\mathbb{R}^d$. In the generic case, rigidity is determined by the generic $d$-dimensional rigidity matroid of $G$. The combinatorial nature of this matroid is well understood when $d=1,2$ but open when $d\geq 3$. Jackson and Jord\'an 2005 characterised independence in this matroid for connected graphs with minimum degree at most $d+1$ and maximum degree at most $d+2$. Their characterisation is known to be false for $(d+2)$-regular graphs when $d\geq 4$ but when $d=3$ it remained open. Indeed they conjectured that their characterisation extends to 5-regular graphs when $d=3$. The purpose of this article is to prove their conjecture. That is, we prove that every 5-regular graph that has at most $3n-6$ edges in any subgraph on $n\geq 3$ vertices is independent in the generic 3-dimensional rigidity matroid.