  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
