On Compatible Normal Odd Partitions in Cubic Graphs

Details On Compatible Normal Odd Partitions in Cubic Graphs On Compatible Normal Odd Partitions in Cubic Graphs

2012-01-27

arxiv.org/abs/1201.5729v1

Computer Science

Discrete Mathematics

Accepted for publication in Journal of Graph Theory

Scientific paper

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

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