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Cyclic orderings and cyclic arboricity of matroids

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We prove a general result concerning cyclic orderings of the elements of a matroid. For each matroid M, weight functionω:E(M)→N, and positive integer D, the following are equivalent. (1) For allA⊆E(M), we have∑a∈Aω(a)D⋅r(A). (2) There is a map ϕ that assigns to each element e ofE(M)a setϕ(e)ofω(e)cyclically consecutive elements in the cycle(1,2,…,D)so that each set{e|i∈ϕ(e)}, fori=1,…,D, is independent. As a first corollary we obtain the following. For each matroid M such that|E(M)|andr(M)are coprime, the following are equivalent. (1) For all non-emptyA⊆E(M), we have|A|/r(A)|E(M)|/r(M). (2) There is a cyclic permutation ofE(M)in which all sets ofr(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.

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