Cayley snarks and almost simple groups

A Cayley snark is a cubic Cayley graph which is not 3-edge-colourable. In t he paper we discuss the problem of the existence of Cayley snarks. This pro blem is closely related to the problem of the existence of non-hamiltonian Cayley graphs and to the question whether every Cayley graph admits a nowhe re-zero 4-flow. So far, no Cayley snarks have been found. On the other hand, we prove that the smallest example of a Cayley snark, if it exists, comes either from a n on-abelian simple group or from a group which has a single non-trivial prop er normal subgroup. The subgroup must have index two and must be either non -abelian simple or the direct product of two isomorphic non-abelian simple groups.