Applications Of Ordinary Voltage Graph Theory To Graph Embeddability, Part 1

TL;DR

This paper explores graph embeddings on surfaces like the torus and Klein bottle using homological invariants, revealing new embedding properties of generalized Petersen graphs and their symmetries.

Contribution

It introduces a matrix invariant for detecting homological invariance and demonstrates novel embedding and symmetry extension results for specific graphs on surfaces.

Findings

Existence of graphs with free group actions extending to surface automorphisms.

Generalized Petersen graphs $GP(2p,2)$ embed in the torus with certain symmetry properties.

Certain Petersen graphs embed in the Klein bottle with symmetry extensions.

Abstract

We study embeddings of a graph $G$ in a surface $S$ by considering representatives of different classes of $H_1(S)$ and their intersections. We construct a matrix invariant that can be used to detect homological invariance of elements of the cycle space of a cellularly embedded graph. We show that: for each positive integer $n$, there is a graph embeddable in the torus such that there is a free $\mathbb{Z}_{2p}$-action on the graph that extends to a cellular automorphism of the torus; for an odd prime $p$ greater than 5 the Generalized Petersen Graphs of the form $GP(2p,2)$ do cellularly embed in the torus, but not in such a way that a free-action of a group on $GP(2p,2)$ extends to a cellular automorphism of the torus; the Generalized Petersen Graph $GP(6,2)$ does embed in the the torus such that a free-action of a group on $GP(6,2)$ extends to a cellular automorphism of the torus; and we show that for any odd $q$, the Generalized Petersen Graph $GP(2q,2)$ does embed in the Klein bottle in such a way that a free-action of a group on the graph extends to a cellular automorphism of the Klein bottle.