The hunting of a snark with total chromatic number 5

Abstract

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 Four-Color 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 ofG, denoted by χT(G), is the least k for which G has a k-total-coloring. Rosenfeld has shown 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’s families and an infinite square-free snark family (including the Loupekhine and Goldberg snarks) is 4. Relaxing any of the conditions of cyclic-edge-connectivity and chromatic index, we exhibit cubic graphs with total chromatic number 5.