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Cyclic orderings and cyclic arboricity of matroids

van den Heuvel, Jan ORCID: 0000-0003-0897-9148 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
Date Deposited: 17 Oct 2011 09:02
Last Modified: 16 May 2024 01:23

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