Cyclic Orderings and Cyclic Arboricity of Matroids

Abstract

We prove a general result concerning cyclic orderings of the elements of a matroid. For each matroid $M$, weight function $\omega :E(M)\to \mathbb{N}$, 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 $\varphi$ that assigns to each element $e$ of $E(M)$ a set $\varphi (e)$ of $\omega (e)$ cyclically consecutive elements in the cycle $(1,2,...,D)$ so that each set $\{e|i\in \varphi (e)\}$, 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.

arXiv comment12 pages