Three-edge-coloring (Tait coloring) cubic graphs on the torus: A proof of Grünbaum's conjecture

Abstract

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the torus, with the exceptional graph class called "Petersen-like", is 3-edge-colorable. This means every (non-trivial) 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 Blanuša snarks; the rest are exposed by Vodopivec in 2008. This proves a strengthening of the well-known, long-standing conjecture of Grünbaum from 1968. This implies that a 2-connected cubic (multi)graph that can be embedded in the torus is not 3-edge-colorable if and only if it can be obtained from a dot product of copies of the Petersen graph by replacing its vertices with 2-edge-connected planar cubic (multi)graphs. Here, replacing a vertex $v$ in a cubic graph $G$ is the operation that takes a 2-connected planar cubic multigraph $H$ and one of its vertices $u$ of degree 3, unifying $G-v$ and $H-u$ and connecting the neighbors of $v$ in $G-v$ with the neighbors of $u$ in $H-u$ with a matching. This result is a highly nontrivial generalization of the Four Color Theorem, and its proof requires a combination of extensive computer verification and computer-free extension of existing proofs on colorability. An important consequence of this result is a very strong version of the Tutte 4-Flow Conjecture for toroidal graphs. We show that a 2-edge connected graph embedded in the torus admits a nowhere-zero 4-flow unless it is Petersen-like (in which case it does not admit nowhere-zero 4-flows). Observe that this is a vast strengthening over the Tutte 4-Flow Conjecture on the torus, which assumes that the graph does not contain the Petersen graph as a minor because almost all toroidal graphs contain the Petersen graph minor, but almost none are Petersen-like.