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 $\omega:E(M)\rightarrow\mathbb{N}$, 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.