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.