Nonplanar graphs and well-covered cycles
If G is planar then the collection \({\mathcal C}\) of face boundaries of a planar embedding satisfy the following properties: (i) each edge of G is in exactly two members of \({\mathcal C}\); and (ii) for each spanning tree T of G, at least two members of \({\mathcal C}\) have all but one edge in T. Rosenfeld conjectured in 1988 that a 2-connected graph with a set of cycles \({\mathcal C}\) satisfying (i) and (ii) is planar. This note furnishes a counterexample to Rosenfeld's conjecture, which is isomorphic to the line graph of the Petersen graph. That it satisfies (ii) is easily shown by looking at its geometric dual.
scientific article; zbMATH DE number 4210173