Minimum degree conditions for graph rigidity

TL;DR

This paper establishes minimum degree thresholds that ensure graph rigidity in high-dimensional spaces, providing tight bounds for small dimensions and approximate bounds for larger ones, with additional insights into graph partitioning properties.

Contribution

It introduces new tight and approximate bounds for minimum degree conditions guaranteeing graph rigidity in various dimensions, advancing understanding in geometric graph theory.

Findings

For small d, minimum degree at least (n+d)/2 - 1 guarantees rigidity.

For larger d, minimum degree at least (n+2d)/2 - 1 guarantees rigidity.

Graphs with minimum degree d have pseudoachromatic number at least d+1.

Abstract

We study minimum degree conditions that guarantee that an $n$-vertex graph is rigid in $\mathbb{R}^d$. For small values of $d$, we obtain a tight bound: for $d = O(\sqrt{n})$, every $n$-vertex graph with minimum degree at least $(n+d)/2 - 1$ is rigid in $\mathbb{R}^d$. For larger values of $d$, we achieve an approximate result: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $(n+2d)/2 - 1$ is rigid in $\mathbb{R}^d$. This bound is tight up to a factor of two in the coefficient of $d$. As a byproduct of our proof, we also obtain the following result, which may be of independent interest: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $d$ has pseudoachromatic number at least $d+1$; namely, the vertex set of such a graph can be partitioned into $d+1$ subsets such that there is at least one edge between each pair of subsets. This is tight.