The Gram dimension of a graph

TL;DR

This paper characterizes graphs with Gram dimension at most 4 by forbidden minors, specifically $K_5$ and $K_{2,2,2}$, connecting to graph realizability concepts.

Contribution

It provides a complete forbidden minor characterization for graphs with Gram dimension at most 4, extending known results for lower dimensions.

Findings

Graphs with Gram dimension ≤ 4 are characterized by the absence of $K_5$ and $K_{2,2,2}$ minors.

The results connect Gram dimension to $d$-realizability of graphs.

Implications for understanding graph embeddings in Euclidean spaces.

Abstract

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner products on the edges of the graph. The class of graphs satisfying $\gd(G) \le k$ is minor closed for fixed $k$, so it can characterized by a finite list of forbidden minors. For $k\le 3$, the only forbidden minor is $K_{k+1}$. We show that a graph has Gram dimension at most 4 if and only if it does not have $K_5$ and $K_{2,2,2}$ as minors. We also show some close connections to the notion of $d$-realizability of graphs. In particular, our result implies the characterization of 3-realizable graphs of Belk and Connelly \cite{Belk,BC}.