On 3-Connected Planar Graphs with Unique Orientable Circuit Double Covers

TL;DR

This paper explores the unique orientable circuit double covers of 3-connected planar graphs, extending previous results from cubic graphs to all such graphs, and relates them to embeddings on orientable surfaces.

Contribution

It generalizes the characterization of graphs with a unique orientable circuit double cover from cubic to all 3-connected planar graphs.

Findings

3-connected planar graphs have a unique orientable circuit double cover if and only if they are duals of Apollonian networks.

The result previously known for cubic graphs is extended to all 3-connected planar graphs.

The work supports the Orientable Strong Embedding Conjecture for a broader class of graphs.

Abstract

A circuit double cover of a bridgeless graph is a collection of even subgraphs such that every edge is contained in exactly two subgraphs of the given collection. Such a circuit double cover describes an embedding of the corresponding graph onto a surface. In this paper, we investigate the well-known Orientable Strong Embedding Conjecture. This conjecture proposes that every bridgeless graph has a circuit double cover describing an embedding on an orientable surface. In a recent paper, we have proved that a 3-connected cubic planar graph G has exactly one orientable circuit double cover if and only if G is the dual graph of an Apollonian network. In this paper, we extend this result by demonstrating that this characterisation applies to any 3-connected planar graph, regardless of whether it is cubic.