Counting matroids in minor-closed classes

Pendavingh, Rudi; Pol, J.G. van der

doi:10.1016/j.jctb.2014.10.001

Cited by 10 publications

(19 citation statements)

References 14 publications

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“…Oxley, Semple, Warshauer, and Welsh show in [OSWW13] that asymptotically almost all matroids are 3-connected. The present authors show in [PvdP15a] that if N is U 2,k , U 3,6 , or one of several other matroids on 6 elements, then almost all matroids have N as a minor, and in [PvdP15b] that almost all matroids on n elements have rank between n/2−β √ n and n/2+β √ n whenever β > ln(2)/2. Given any matroid M on E of rank r, the set of bases B of M consists of subsets of E of cardinality r, so that B ⊆ V (G).…”

Section: Introductionmentioning

confidence: 56%

On the Number of Bases of Almost All Matroids

Pendavingh

Pol

2018

Combinatorica

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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 thatfor 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 ≥ Ω( log(n)), and (4) do not arise as the truncation of another matroid.Our argument is based on a refined 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.Conjecture 1.1. As n → ∞, asymptotically almost all matroids on n elements are sparse paving.If this conjecture were true, then several other asymptotic properties of matroids would follow with little extra work, as it would suffice to establish the property for almost all sparse paving matroids. For example, almost all sparse paving matroids contain a fixed uniform matroid, are highly connected, have high girth, and the analogous statements for general matroids would follow. We next describe our intuition supporting Conjecture 1.1, which we partially substantiated to obtain the results of this paper.Consider the Johnson graph J(E, r), whose vertices are the subsets of E of cardinality r, and in which two vertices X, Y are adjacent exactly if |X ∩ Y | = r − 1. The Johnson graph will serve as an 'ambient space' for all the matroids on ground set E and of rank r. In what follows, we will write G := J(E, r).

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Section: Introductionmentioning

confidence: 56%

On the Number of Bases of Almost All Matroids

Pendavingh

Pol

2018

Combinatorica

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“…In [23] two of the current authors show that this conjecture holds for each of the matroids N = U 2,k , U 3,6 , P 6 , Q 6 , and R 6 , by deriving bounds on the cover complexity of matroids that do not have such a minor.…”

Section: Conjecture 22mentioning

confidence: 87%

On the number of matroids

2014

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We consider the problem of determining mn, the number of matroids on n elements. The best known lower bound on mn is due to Knuth (1974) who showed that log log mn is at least n− 3 2 log n−O(1). On the other hand, Piff (1973) showed that log log mn ≤ n−log n+log log n+O (1), and it has been conjectured since that the right answer is perhaps closer to Knuth's bound.We show that this is indeed the case, and prove an upper bound on log log mn that is within an additive 1 + o(1) term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of stable sets in the Johnson graph to give a compressed representation of matroids.

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“…This builds upon work by Pendavingh and van der Pol [24] and we are able to resolve the question for some specific matroids as well as classes of matroids. Restricting to matroids of fixed rank, even more extensive results are possible.…”

Section: Chapter 1 Introductionmentioning

confidence: 92%

“…Some progress on asymptotic questions has been made recently, especially by Pendavingh and van der Pol who showed (amongst other things) that asymptotically almost all matroids are k-connected [23], are not a truncation [23], contain a large uniform minor [23], and contain minors isomorphic to each of a small collection of matroids [24].…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

“…The only exception to this is a few preliminary lemmas that were so fundamental and intuitive that finding an attribution seemed impossible; in these cases we have generally included our own improvised proof for completeness. Of the non-original work presented, the most important sources are Pendavingh and van der Pol [24] [23] [25], followed by Mayhew, Newman, Welsh and Whittle [17] [20].…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

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Randomness in classes of matroids

Critchlow¹

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<p>This thesis is inspired by the observation that we have no good random model for matroids. That stands in contrast to graphs, which admit a number of elegant random models. As a result we have relatively little understanding of the properties of a "typical" matroid. Acknowledging the difficulty of the general case, we attempt to gain a grasp on randomness in some chosen classes of matroids. Firstly, we consider sparse paving matroids, which are conjectured to dominate the class of matroids (that is to say, asymptotically almost all matroids would be sparse paving). If this conjecture were true, then many results on properties of the random sparse paving matroid would also hold for the random matroid. Sparse paving matroids have limited richness of structure, making counting arguments in particular more feasible than for general matroids. This enables us to prove a number of asymptotic results, particularly with regards to minors. Secondly, we look at Graham-Sloane matroids, a special subset of sparse paving matroids with even more limited structure - in fact Graham-Sloane matroids on a labelled groundset can be randomly generated by a process as simple as independently including certain bases with probability 0.5. Notably, counting Graham-Sloane matroids has provided the best known lower bound on the total number of matroids, to log-log level. Despite the vast size of the class we are able to prove severe restrictions on what minors of Graham-Sloane matroids can look like. Finally we consider transversal matroids, which arise naturally in the context of other mathematical objects - in particular partial transversals of set systems and partial matchings in bipartite graphs. Although transversal matroids are not in one-to-one correspondence with bipartite graphs, we shall link the two closely enough to gain some useful results through exploiting the properties of random bipartite graphs. Returning to the theme of matroid minors, we prove that the class of transversal matroids of given rank is defined by finitely many excluded series-minors. Lastly we consider some other topics, including the axiomatisability of transversal matroids and related classes.</p>

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Counting matroids in minor-closed classes

Cited by 10 publications

References 14 publications

On the Number of Bases of Almost All Matroids

On the Number of Bases of Almost All Matroids

On the number of matroids

Randomness in classes of matroids

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