Reinforcing a Matroid to Have k Disjoint Bases

Hong-Jian Lai,Jinquan Xu,Ping Li,Yanting Liang

doi:10.4236/am.2010.13030

Hong-Jian Lai, Jinquan Xu + Show 2 more

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https://doi.org/10.4236/am.2010.13030

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Abstract

Let denote the maximum number of disjoint bases in a matroid . For a connected graph , let , where is the cycle matroid of . The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with . Edmonds generalizes this theorem to matroids. In [1] and [2], for a matroid with , elements with the property that have been characterized in terms of matroid invariants such as strength and -partitions. In this paper, we consider matroids with , and determine the minimum of , where is a matroid that contains as a restriction with both and . This minimum is expressed as a function of certain invariants of , as well as a min-max formula. These are applied to imply former results of Haas [3] and of Liu et al. [4].

Highlights

For a matroid M, let (M ) denote the maximum number of disjoint bases of M
Theorem 4.2 (Haas, Theorem 1 of [3]) The following are equivalent for a graph G, and integers k > 0 and l>0. 1) | E(G) |= k(| V (G) | 1) l and for subgraphs H of G with at least 2 vertices, | E(H ) | k(| V (H ) | 1) . 2) There exists some l edges which when added to G result in a graph that can be decomposed into k spanning trees
It follows by the assumption that | E(G) |= k(| V (G) | 1) l and by Lemma 3.1 2) that

Summary

Introduction

For a matroid M , let (M ) denote the maximum number of disjoint bases of M. Theorem 1.2 (Edmonds [11]) Let M be a matroid with r(M ) > 0. 1) There exist an integer m N , and an m -tuple (l1,l2 ,...,lm ) of rational numbers in Q such that. Such that for each i with 1 i m , M | Ji is an maximal restriction of M with (M | Ji ) = li. For a matroid M , the m -tuple (l1,l2 ,...,lm ) and the sequence in (4) will be referred as the -spectrum and the -decomposition of M , respectively. We shall present some of the useful properties related to the strength and the fractional arboricity of a matroid M , and to the decomposition of M. We shall show some applications of our main results

Preliminaries

Applications