On 3-Connected Planar Graphs with Unique Orientable Circuit Double Covers
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.
Cite
@article{arxiv.2601.10171,
title = {On 3-Connected Planar Graphs with Unique Orientable Circuit Double Covers},
author = {Meike Weiß and Reymond Akpanya and Alice C. Niemeyer},
journal= {arXiv preprint arXiv:2601.10171},
year = {2026}
}