Cyclic Orderings and Cyclic Arboricity of Matroids

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 $\sum_{a\in A}\omega(a)\le D\cdot r(A)$\mathrm{.}(2)Thereisamap$\phi$thatassignstoeachelement$e$of$E(M)$aset$\phi(e)$of$\omega(e)$ cyclically consecutive elements in the cycle $(1,2,\mathrm{.}\mathrm{.}\mathrm{.},D)$ so that each set $\{e\mathrm{\mid }i\in \varphi (e)\}$, for $i=1,\mathrm{.}\mathrm{.}\mathrm{.},D$, is independent. As a first corollary we obtain the following. For each matroid $M$ so that $\mathrm{\mid }E(M)\mathrm{\mid }$ and $r(M)$ are coprime, the following are equivalent. (1) For all non-empty $A\subseteq E(M)$, we have $\mathrm{\mid }A\mathrm{\mid }\mathrm{/}r(A)\le \mathrm{\mid }E(M)\mathrm{\mid }\mathrm{/}r(M)$. (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.