Primary Non-QE Graphs on Six Vertices

TL;DR

This paper classifies all six-vertex graphs into QE, non-QE, and primary non-QE categories, providing a complete understanding of their geometric embedding properties.

Contribution

It offers the first complete classification of six-vertex graphs based on their quadratic embedding characteristics, introducing the concept of primary non-QE graphs.

Findings

Complete classification of six-vertex graphs into QE, non-QE, and primary non-QE categories

Identification of primary non-QE graphs as minimal non-QE graphs without non-QE subgraphs

Clarification of the structure of graphs related to quadratic embeddings in Euclidean space

Abstract

A connected graph is called of non-QE class if it does not admit a quadratic embedding in a Euclidean space. A non-QE graph is called primary if it does not contain a non-QE graph as an isometrically embedded proper subgraph. The graphs on six vertices are completely classified into the classes of QE graphs, of non-QE graphs, and of primary non-QE graphs.