Three-Edge-Coloring Projective Planar Cubic Graphs: A Generalization of the Four Color Theorem

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 (nontrivial) 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. Here, a replacement of a vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$v$</tex> in a cubic graph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> is the operation that takes a 2-connected planar (cubic) multigraph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex> containing some vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$u$</tex> of degree 3, unifying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G-v$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H-u$</tex>, and connecting the vertices in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N_{G}[v]$</tex> in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G-v$</tex> with the three neighbors of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$u$</tex> in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H-u$</tex> with 3 edges. Any graph obtained in such a way is said to be Petersen-like. 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. Using this result, we obtain the following algorithmic consequence. Input: A cubic graph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex>. Output: Either a 3-edge-coloring of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex>, an obstruction showing that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> is not 3-edge-colorable, or the conclusion that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> cannot be embedded in the projective plane (certified by exposing a forbidden minor for the projective plane contained in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex>). Time complexity: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(n^{2})$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n=\vert V(G)\vert$</tex>. 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 Petersen-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 5000 reducible configurations, are available on Github<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup>https://github.com/edge-coloring. Refer to the “README.md” file in each directory for instructions on how to run each program. which can be considered as an addendum to this paper. Moreover, we provide pseudocodes for all our computer verifications.