Three-edge-coloring (Tait coloring) cubic graphs and nowhere-zero 4-flow for graphs on the torus

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the torus, with the exception of two specific infinite families of “Petersen-like” graphs, is 3-edge-colorable. This shows that every toroidal snark can be obtained from several copies of the Petersen graph using the dot product operation. The first two snarks in this family are the Petersen graph and one of the Blanuša snarks; the rest were exposed by Belcastro and Kaminski and by Vodopivec. This proves a strengthening of the well-known, long-standing conjecture of Grünbaum from 1968.