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Wagner’s theorem states that a graph is planar (i.e., it can be embedded in the real plane without crossing edges) iff it contains neither K5 nor K3,3 as a minor. We provide a combinatorial representation of embeddings in the plane that abstracts from topological properties of plane embeddings (e.g., angles or distances), representing only the combinatorial properties (e.g., arities of faces or the clockwise order of the outgoing edges of a vertex). The representation employs combinatorial hypermaps as used by Gonthier in the proof of the four-color theorem. We then give a formal proof that for every simple graph containing neither K5 nor K3,3 as a minor, there exists such a combinatorial plane embedding. Together with the formal proof of the four-color theorem, we obtain a formal proof that all graphs without K5 and K3,3 minors are four-colorable. The development is carried out in Coq, building on the mathematical components library, the formal proof of the four-color theorem, and a general-purpose graph library developed previously. 2012 ACM Subject Classification Mathematics of computing → Graphs and surfaces; Theory of computation → Logic and verification

A Variant of Wagner's Theorem Based on Combinatorial Hypermaps