van den Heuvel, Jan 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 (Other) |
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Official URL: | http://arxiv.org/ |

Additional Information: | © 2009 The authors |

Library of Congress subject classification: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Rights: | http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx |

Date Deposited: | 09 Apr 2010 13:15 |

URL: | http://eprints.lse.ac.uk/27681/ |

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