Cyclic Orderings and Cyclic Arboricity of Matroids
Jan van den Heuvel, Stéphan Thomassé · arXiv (Cornell University) · 2009
We derive a general result concerning cyclic orderings of the elements of a matroid. As corollaries we obtain two further results. The first corollary proves a conjecture of Gonçalves [7], stating that the circular arboricity of a matroid is equal to its fractional arboricity. This generalises a well-known result from Nash-Williams on covering graphs by spanning trees, and a result from Edmonds on covering matroids by bases. The second corollary is the proof of a special case of a conjecture of Kajitani et al [8] about the possibility of ordering the elements of a matroid of rank r to have a cyclic ordering in which every r consecutive elements form a base of the matroid.