Three-edge-coloring projective planar cubic graphs: A generalization of the Four Color Theorem

TL;DR

This paper proves that all cyclically 4-edge-connected cubic graphs embeddable in the projective plane, except the Petersen graph, are 3-edge-colorable, extending the Four Color Theorem and exploring related flow and coloring dualities.

Contribution

It generalizes the Four Color Theorem to projective planar cubic graphs and establishes a coloring-flow duality, with extensive computer-assisted proofs and publicly available code.

Findings

Petersen graph is the only non-trivial snark embeddable in the projective plane.

A duality between 3-edge-colorability and 5-vertex-colorability in the projective plane.

Strengthening of the Tutte 4-flow conjecture for graphs on the projective plane.

Abstract

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the projective plane, with the single exception of the Petersen graph, is 3-edge-colorable. In other words, the only (non-trivial) snark that can be embedded in the projective plane is the Petersen graph. This implies that a 2-connected cubic (multi)graph that can be embedded in the projective plane is not 3-edge-colorable if and only if it can be obtained from the Petersen graph by replacing each vertex by a 2-edge-connected planar cubic (multi)graph. This result is a 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 unexpected consequence of this result is a coloring-flow duality statement for the projective plane: A cubic graph embedded in the projective plane is 3-edge-colorable if and only if its dual multigraph is 5-vertex-colorable. Moreover, we show that a 2-edge connected graph embedded in the projective plane admits a nowhere-zero 4-flow unless it is Peteren-like (in which case it does not admit nowhere-zero 4-flows). This proves a strengthening of the Tutte 4-flow conjecture for graphs on the projective plane. Some of our proofs require extensive computer verification. The necessary source codes, together with the input and output files and the complete set of more than 6000 reducible configurations are available on Github (https://github.com/edge-coloring) which can be considered as an Addendum to this paper. Moreover, we provide pseudocodes for all our computer verifications.