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On the Number of Bases of Almost All Matroids

On the Number of Bases of Almost All Matroids For a matroid M of rank r on n elements, let b(M) denote the fraction of bases of M among the subsets of the ground set with cardinality r. We show that $$\Omega \left( {1/n} \right) \leqslant 1 - b\left( M \right) \leqslant O\left( {\log {{\left( n \right)}^3}/n} \right)a\;sn \to \infty $$ Ω ( 1 / n ) ≤ 1 − b ( M ) ≤ O ( log ( n ) 3 / n ) a s n → ∞ for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a U k,2k -minor, whenever k≤O(log(n)), (2) have girth ≥Ω(log(n)), (3) have Tutte connectivity $$\geq\Omega\;{(\sqrt {log(n)})}$$ ≥ Ω ( l o g ( n ) ) , and (4) do not arise as the truncation of another matroid. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Combinatorica Springer Journals

On the Number of Bases of Almost All Matroids

Pendavingh, Rudi; van der Pol, Jorn
Combinatorica , Volume 38 (4) – Jun 5, 2018
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31 pages

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Publisher
Springer Journals
Copyright
Copyright © 2018 by János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Combinatorics; Mathematics, general
ISSN
0209-9683
eISSN
1439-6912
DOI
10.1007/s00493-016-3594-4
Publisher site
See Article on Publisher Site

Abstract

For a matroid M of rank r on n elements, let b(M) denote the fraction of bases of M among the subsets of the ground set with cardinality r. We show that $$\Omega \left( {1/n} \right) \leqslant 1 - b\left( M \right) \leqslant O\left( {\log {{\left( n \right)}^3}/n} \right)a\;sn \to \infty $$ Ω ( 1 / n ) ≤ 1 − b ( M ) ≤ O ( log ( n ) 3 / n ) a s n → ∞ for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a U k,2k -minor, whenever k≤O(log(n)), (2) have girth ≥Ω(log(n)), (3) have Tutte connectivity $$\geq\Omega\;{(\sqrt {log(n)})}$$ ≥ Ω ( l o g ( n ) ) , and (4) do not arise as the truncation of another matroid.

Journal

Combinatorica – Springer Journals

Published: Jun 5, 2018

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