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Cyclic orderings and cyclic arboricity of matroids

van den Heuvel, Jan and Thomassé, Stéphane (2012) Cyclic orderings and cyclic arboricity of matroids. Journal of Combinatorial Theory, Series B, 102 (3). pp. 638-646. ISSN 0095-8956

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Identification Number: 10.1016/j.jctb.2011.08.004


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.

Item Type: Article
Official URL:
Additional Information: © 2012 Elsevier
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 17 Oct 2011 09:02
Last Modified: 20 Apr 2021 00:54

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