Edge-coloring 4- and 5-regular projective planar graphs with no Petersen-minor

An $r$-regular graph is an $r$-graph, if every odd set of vertices is connected to its complement by at least $r$ edges. We prove for $r \in {4,5}$, every projective planar $r$-graph with no Petersen-minor is $r$-edge colorable.

💡 Research Summary

The paper investigates edge‑coloring of regular graphs embedded in the projective plane, focusing on the case where the graph is r‑regular with r ∈ {4,5} and contains no Petersen minor. An r‑regular graph that satisfies the condition that every odd vertex set is incident with at least r edges across its cut is called an r‑graph. This property is closely related to Tutte’s 4‑flow conjecture and to Seymour’s broader conjecture that any r‑graph without a Petersen minor should be class 1 (i.e., its edge‑chromatic number equals its maximum degree).

The authors first collect necessary terminology and known results. They recall that planar graphs are automatically Petersen‑minor‑free, so planar r‑graphs with r ≤ 8 are already known to be class 1. Recent work by Inoue et al. showed that the only snark (a cubic class 2 graph) embeddable in the projective plane is the Petersen graph itself. This motivates the study of higher‑degree regular graphs on the projective plane.

A minimal counterexample H is defined with respect to vertex count, number of 3‑edges, and number of 2‑edges. The paper proves a series of structural properties of such a minimal H: it must be non‑planar, 3‑connected, have no non‑trivial tight cuts, satisfy |∂H(X)| ≥ 4 for every vertex set X, and have maximum multiplicity µ(H) ≤ r − 2. These constraints force H to have a circular 2‑cell embedding where every facial walk is a simple circuit.

For the case r = 4, the authors employ the discharging method. Each face f receives an initial charge ω(f)=d(f)−4, where d(f) is the length of the facial walk. Euler’s formula yields a total initial charge of –4. Using Lemma 1.13 and Corollary 2.3 they show that any face adjacent to a small face (size ≤ 3) must be large (size ≥ 5). Two redistribution rules are introduced: (R1) each 2‑face receives one unit from each adjacent face, and (R2) each 3‑face receives one‑third from each adjacent face. Careful analysis shows that after redistribution every face ends with non‑negative charge, contradicting the total –4 sum. Hence no minimal counterexample exists and every projective‑planar 4‑graph without a Petersen minor is class 1.

The r = 5 case builds on the r = 4 result. Assuming a minimal counterexample G, the authors examine the structure of faces and introduce the concepts of strong e‑colorings, mates, and bad triangles. Lemma 1.11 guarantees that for any edge e, each of the r T‑joins in a strong e‑coloring has a mate. Lemma 1.12 ensures mates can be chosen without creating bad triangles. Using these tools they construct strong colorings for edges that belong to multiple parallel edges (up to r − 2 of them). After removing the parallel edges and suppressing the incident vertices, the resulting graph G′ is a 4‑regular projective‑planar r‑graph without a Petersen minor, which by the previous section is class 1. The coloring of G′ can be lifted back to G, and a discharging argument analogous to the 4‑regular case (with adjusted charge transfers) shows that every face again ends with non‑negative charge, contradicting the global charge deficit.

Consequently, the main theorem is proved: any projective‑planar r‑graph with r ≤ 5 that does not contain a Petersen minor is class 1. The paper concludes by noting that this settles Seymour’s conjecture for the projective plane up to degree five, discusses possible extensions to higher degrees and to other non‑orientable surfaces, and highlights the utility of strong e‑colorings together with discharging in handling multigraphs on non‑planar surfaces.