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

TL;DR

The paper proves that the Petersen graph is the sole non-$3$-edge-colorable graph among bridgeless cubic graphs embeddable in the projective plane, up to Petersen-like substitutions that replace vertices by planar cubic graphs. It extends the Four Color Theorem methodology to the projective plane by building an enormous unavoidable set of reducible configurations and a comprehensive discharging framework with 169 rules, augmented by extensive computer verification and computer-free extensions. A surprising corollary is a coloring-flow duality on the projective plane: a cubic graph embedded in the projective plane is $3$-edge-colorable iff its dual is vertex $5$-colorable, strengthening Tutte’s $4$-flow conjecture in this setting. The work yields a quadratic-time algorithm that either outputs a $3$-edge-coloring, or an obstruction, or certifies non-embeddability, thereby providing a practical decision procedure for projective-planar cubic graphs and advancing understanding of colorability and flows on nonorientable surfaces.

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.

Figures (38)

• Figure 1: The first reducible configuration that consists of four vertices of degree five in the dual graph. The bold lines represent the contraction (which corresponds to edge-deletion in the cubic graph) used for C-reducibility.
• Figure 2:
• Figure 3: Shapes used to designate vertices of specific degrees.
• Figure 4: The eight cases that remain after computer-check in the proof of Lemma \ref{['pairs6']}. The dotted line represents edges when we assume two vertices of distance five are adjacent.
• Figure 5: Two C-reducible islands in $\Delta^6$, having contraction size $4$. The contraction edges are depicted by dotted lines and the non-contractible curve is represented with a blue curved line.

Theorems & Definitions (134)

• Theorem 1.1
• Conjecture 1: Tutte, tutte
• Conjecture 2: Grünbaum grunbaum, 1968
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• proof
• Theorem 1.6
• Definition 2.1