Cubic torus obstructions of small Betti number

TL;DR

This paper provides a theoretical classification of cubic graphs that cannot be embedded into a torus, focusing on those with Betti number up to eight, advancing understanding of topological obstructions.

Contribution

It offers the first comprehensive classification of cubic torus obstructions with low Betti number, filling a gap in topological graph theory.

Findings

Classified all cubic torus obstructions with Betti number ≤ 8

Identified key structural properties of these obstructions

Enhanced understanding of graph embeddability in toroidal surfaces

Abstract

The embeddability of graphs into surfaces has been studied for nearly a century. While the complete set of topological obstructions is known for the sphere and the real projective plane, there are only partial results for the torus. Here we present a theoretical classification of cubic torus obstructions with Betti number at most eight.