Relating Embedding and Coloring Properties of Snarks

In 1969, Grünbaum conjectured that snarks do not have polyhedral embeddings into orientable surfaces. To describe the deviation from polyhedrality, we define the defect of a graph and use it to study embeddings of superpositions of cubic graphs into orientable surfaces. Superposition was introduced in [4] to construct snarks with arbitrary large girth. It is shown that snarks constructed in [4] do not have polyhedral embeddings into orientable surfaces. For each k ≥ 2 we construct infinitely many snarks with defect precisely k. We then relate the defect with the resistance r(G) of a cubic graph G which is the size of a minimum color class of a 4-edge-coloring of G. These results are then extended to deal with some weaker versions of the Grünbaum Conjecture.