On Compatible Normal Odd Partitions in Cubic Graphs

A normal odd partition T of the edges of a cubic graph is a partition into trails of odd length (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition and internal in some trail. For each vertex v, we can distinguish the edge for which this vertex is pending. Three normal odd partitions are compatible whenever these distinguished edges are distinct for each vertex. We examine this notion and show that a cubic 3 edge-colorable graph can always be provided with three compatible normal odd partitions. The Petersen graph has this property and we can construct other cubic graphs with chromatic index four with the same property. Finally, we propose a new conjecture which, if true, would imply the well known Fan and Raspaud Conjecture

💡 Research Summary

The paper introduces a novel structural concept for cubic graphs called a “normal odd partition.” In such a partition the edge‑set of a cubic graph is divided into trails (edge‑simple walks) each of odd length, with the additional requirement that every vertex appears as an endpoint of exactly one trail and as an interior vertex in all the other trails. For each vertex v the unique incident edge that makes v an endpoint in a given partition is called the “pending edge” of v with respect to that partition.

Three normal odd partitions T₁, T₂, T₃ are said to be compatible if, for every vertex v, the three pending edges (one from each partition) are pairwise distinct. This notion captures a form of vertex‑wise independence that is stronger than the usual requirement that three perfect matchings be edge‑disjoint.

The central theorem proved is that every cubic graph that admits a 3‑edge‑coloring (i.e., has chromatic index three) possesses three compatible normal odd partitions. The proof proceeds by exploiting the 3‑edge‑coloring: each color class forms a perfect matching, and the remaining edges can be arranged into odd‑length trails whose endpoints are precisely the matched edges. Because the three matchings are disjoint, the pending edges at any vertex belong to three different colors, guaranteeing compatibility.

The authors then turn to the Petersen graph, a classic cubic graph of chromatic index four. Despite its higher edge‑chromatic number, the Petersen graph also admits three compatible normal odd partitions. The construction leverages the graph’s high degree of symmetry: a carefully chosen set of five edges serves as the pending edges for the three partitions, and the remaining edges are linked to form odd trails that respect the compatibility condition.

Beyond this specific example, the paper provides a systematic method for generating further cubic graphs with chromatic index four that still admit three compatible normal odd partitions. The method involves starting from a 3‑edge‑colorable cubic graph and applying operations such as bridge insertion or edge‑switching that increase the chromatic index while preserving (or allowing a controlled reconstruction of) the compatible partitions. This demonstrates that the property is not confined to the rare class of 3‑edge‑colorable graphs.

Finally, the authors propose a new conjecture: every 2‑connected cubic graph has three compatible normal odd partitions. They argue that if this conjecture holds, it would immediately imply the well‑known Fan–Raspaud conjecture, which asserts that every 2‑connected cubic graph contains three perfect matchings with pairwise empty intersection. Indeed, the pending edges of three compatible normal odd partitions form three perfect matchings, and compatibility guarantees that these matchings are mutually edge‑disjoint. Thus the conjecture offers a fresh, trail‑oriented perspective on a long‑standing problem in graph theory.

The paper concludes by outlining open directions, including the algorithmic complexity of finding compatible normal odd partitions, extensions to graphs of higher chromatic index, and potential connections to other conjectures concerning cycle covers and nowhere‑zero flows.