van den Heuvel, Jan 
ORCID: 0000-0003-0897-9148 and Thomassé, Stéphane
(2009)
Cyclic orderings and cyclic arboricity of matroids.
.
arXiv.
Abstract
We prove a general result concerning cyclic orderings of the elements of a matroid. For each matroid $M$, weight function $\omega:E(M)\rightarrow\mathbb{N}$, and positive integer $D$, the following are equivalent. (1) For all $A\subseteq E(M)$, we have $\sum_{a\in A}\omega(a)\le D\cdot r(A)$. (2) There is a map $\phi$ that assigns to each element $e$ of $E(M)$ a set $\phi(e)$ of $\omega(e)$ cyclically consecutive elements in the cycle $(1,2,...,D)$ so that each set $\{e|i\in\phi(e)\}$, for $i=1,...,D$, is independent. As a first corollary we obtain the following. For each matroid $M$ so that $|E(M)|$ and $r(M)$ are coprime, the following are equivalent. (1) For all non-empty $A\subseteq E(M)$, we have $|A|/r(A)\le|E(M)|/r(M)$. (2) There is a cyclic permutation of $E(M)$ in which all sets of $r(M)$ cyclically consecutive elements are bases of $M$. A second corollary is that the circular arboricity of a matroid is equal to its fractional arboricity. These results generalise classical results of Edmonds, Nash-Williams and Tutte on covering and packing matroids by bases and graphs by spanning trees.
| Item Type: | Monograph (Report) |
|---|---|
| Official URL: | http://arxiv.org/ |
| Additional Information: | © 2009 The authors |
| Divisions: | Mathematics |
| Subjects: | Q Science > QA Mathematics |
| Date Deposited: | 09 Apr 2010 13:15 |
| Last Modified: | 01 Apr 2024 07:50 |
| URI: | http://eprints.lse.ac.uk/id/eprint/27681 |
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