Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic

TL;DR

This paper characterizes embeddings of 3-connected 3-regular planar graphs on surfaces with non-negative Euler characteristic, establishing a correspondence with subgraphs of the dual on the sphere, and provides algorithms for enumeration.

Contribution

It introduces a novel characterization of embeddings on various surfaces and links them to dual subgraphs, enabling explicit bounds and enumeration algorithms.

Findings

Established a one-to-one correspondence between embeddings and dual subgraphs.

Provided explicit bounds for the number of inequivalent embeddings.

Developed algorithms for enumerating and counting embeddings.

Abstract

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.