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

arXiv (Cornell University)

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.

Authors 3

1. Meike Weiß Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University , Aachen , Germany

2. Reymond Akpanya

   The University of Sydney

   Affiliation as printed

   The University of Sydney , Sydney , Australia

3. Alice C. Niemeyer Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University , Germany