On Compatible Normal Odd Partitions in Cubic Graphs

TL;DR

This paper investigates compatible normal odd partitions in cubic graphs, demonstrating their existence in 3-edge-colorable graphs and constructing examples in non-colorable graphs, proposing a conjecture related to the Fan-Raspaud Conjecture.

Contribution

It introduces the concept of compatible normal odd partitions in cubic graphs and proves their existence in 3-edge-colorable graphs, also proposing a new conjecture linked to a famous conjecture.

Findings

Cubic 3-edge-colorable graphs always have three compatible normal odd partitions.

The Petersen graph possesses this property.

Constructed cubic graphs with chromatic index four also have compatible normal odd partitions.

Abstract

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