The hunting of a snark with total chromatic number 5

A snark is a cyclically-4-edge-connected cubic graph with chromatic index 4. In 1880, Tait proved that the Four-Color Conjecture is equivalent to the statement that every planar bridgeless cubic graph has chromatic index 3. The search for counter-examples to the FourColor Conjecture motivated the definition of the snarks. A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent or incident elements have different colors. The total chromatic number χT of G is the least k for which G has a k-total-coloring. It is known that the total chromatic number of a cubic graph is either 4 or 5. However, the problem of determining the total chromatic number of a graph is NP-hard even for cubic bipartite graphs. In 2003, Cavicchioli et al. reported that their extensive computer study of snarks shows that all square-free snarks with less than 30 vertices have total chromatic number 4, and asked for the smallest order of a square-free snark with total chromatic number 5. In this paper we prove that the total chromatic number of both Blanuša families and an infinite snark family (including the Loupekhine and Goldberg snarks) is 4. Relaxing any of the conditions of cyclicedge-connectivity and chromatic index, we exhibit cubic graphs with total chromatic number 5.