On Compatible Normal Odd Partitions in Cubic Graphs
Article first published online: 26 JUN 2012
© 2012 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 72, Issue 4, pages 440–461, April 2013
How to Cite
Fouquet, J.-L. and Vanherpe, J.-M. (2013), On Compatible Normal Odd Partitions in Cubic Graphs. J. Graph Theory, 72: 440–461. doi: 10.1002/jgt.21652
- Issue published online: 8 FEB 2013
- Article first published online: 26 JUN 2012
- Manuscript Revised: 20 JAN 2012
- Manuscript Received: 29 MAR 2009
- cubic graph;
A normal odd partition 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.