On the Number of Bases of Almost All Matroids

On the Number of Bases of Almost All Matroids Combinatorica 31pp. COMBINATORICA DOI: 10.1007/s00493-016-3594-4 Bolyai Society { Springer-Verlag RUDI PENDAVINGH, JORN VAN DER POL Received March 4, 2016 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 (1=n)  1 b(M )  O(log(n) =n) as n ! 1 for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a U -minor, whenever kO(log(n)), (2) have k;2k girth (log(n)), (3) have Tutte connectivity ( log(n)), and (4) do not arise as the truncation of another matroid. Our argument is based on a re ned method for writing compressed descriptions of any given matroid, which allows bounding the number of matroids in a class relative to the number of sparse paving matroids. 1. Introduction 1.1. Matroid asymptotics This paper is concerned with the properties that are satis ed by most ma- troids as the size of their ground set tends to in nity. Precisely, for a matroid property P we consider the asymptotic fraction jfM 2 M : M has property Pgj jM j http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Combinatorica Springer Journals

On the Number of Bases of Almost All Matroids

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Publisher
Springer Berlin Heidelberg
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
D.O.I.
10.1007/s00493-016-3594-4
Publisher site
See Article on Publisher Site

Abstract

Combinatorica 31pp. COMBINATORICA DOI: 10.1007/s00493-016-3594-4 Bolyai Society { Springer-Verlag RUDI PENDAVINGH, JORN VAN DER POL Received March 4, 2016 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 (1=n)  1 b(M )  O(log(n) =n) as n ! 1 for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a U -minor, whenever kO(log(n)), (2) have k;2k girth (log(n)), (3) have Tutte connectivity ( log(n)), and (4) do not arise as the truncation of another matroid. Our argument is based on a re ned method for writing compressed descriptions of any given matroid, which allows bounding the number of matroids in a class relative to the number of sparse paving matroids. 1. Introduction 1.1. Matroid asymptotics This paper is concerned with the properties that are satis ed by most ma- troids as the size of their ground set tends to in nity. Precisely, for a matroid property P we consider the asymptotic fraction jfM 2 M : M has property Pgj jM j

Journal

CombinatoricaSpringer Journals

Published: Jun 5, 2018

References

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