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Mapping Planar Graphs into Projective Cubes



Projective cubes are obtained by identifying antipodal vertices of hypercubes. We introduce a general problem of mapping planar graphs into projective cubes. This question, surprisingly, captures several well-known theorems and conjectures in the theory of planar graphs. As a special case , we prove that the Clebsch graph, a triangle-free graph on 16 vertices, is the smallest triangle-free graph to which every triangle-free planar graph admits a homomorphism.