Minor-closed Classes Of Matroids Research Articles

Fractal classes of matroids

A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three. The minor-closure of the class of spikes with at most k circuit-hyperplanes (with k≥5) satisfies a strictly weaker condition: the number of 2t-element matroids in the class is dominated by the number of 2t-element excluded minors. However, there are only finitely many excluded minors with ground sets of odd size.

Projective geometries in exponentially dense matroids. I

We show for each positive integer a that, if M is a minor-closed class of matroids not containing all rank-(a+1) uniform matroids, then there exists an integer n such that either every rank-r matroid in M can be covered by at most rn sets of rank at most a, or M contains the GF(q)-representable matroids for some prime power q and every rank-r matroid in M can be covered by at most rnqr sets of rank at most a. This determines the maximum density of the matroids in M up to a polynomial factor.

Projective geometries in exponentially dense matroids. II

We show for each positive integer a that, if M is a minor-closed class of matroids not containing all rank-(a+1) uniform matroids, then there exists an integer c such that either every rank-r matroid in M can be covered by at most rc rank-a sets, or M contains the GF(q)-representable matroids for some prime power q and every rank-r matroid in M can be covered by at most cqr rank-a sets. In the latter case, this determines the maximum density of matroids in M up to a constant factor.

On minor-closed classes of matroids with exponential growth rate

Let M be a minor-closed class of matroids that does not contain arbitrarily long lines. The growth rate function, h:N→N of M is given byh(n)=max{|M|:M∈M is simple, and r(M)⩽n}. The Growth Rate Theorem shows that there is an integer c such that either: h(n)⩽cn, or (n+12)⩽h(n)⩽cn2, or there is a prime-power q such that qn−1q−1⩽h(n)⩽cqn; this separates classes into those of linear density, quadratic density, and base-q exponential density. For classes of base-q exponential density that contain no (q2+1)-point line, we prove that h(n)=qn−1q−1 for all sufficiently large n. We also prove that, for classes of base-q exponential density that contain no (q2+q+1)-point line, there exists k∈N such that h(n)=qn+k−1q−1−qq2k−1q2−1 for all sufficiently large n.

When excluding one matroid prevents infinite antichains

Geelen, Gerards, and Whittle have announced that there are no infinite sets of binary matroids none of which is isomorphic to a minor of another. In this paper, we use this result to determine precisely when a minor-closed class of matroids with a single excluded minor does not contain such an infinite antichain.

Parametrized Tutte Polynomials of Graphs and Matroids

We generalize and unify results on parametrized and coloured Tutte polynomials of graphs and matroids due to Zaslavsky, and Bollobas and Riordan. We give a generalized Zaslavsky–Bollobas–Riordan theorem that characterizes parametrized contraction–deletion functions on minor-closed classes of matroids, as well as the modifications necessary to apply the discussion to classes of graphs. In general, these parametrized Tutte polynomials do not satisfy analogues of all the familiar properties of the classical Tutte polynomial. We give conditions under which they do satisfy corank-nullity formulas, and also conditions under which they reflect the structure of series-parallel connections.

Strong Tutte functions of matroids and graphs

A strong Tutte function of matroids is a function of finite matroids which satisfies F ( M 1 ⊕ M 2 ) = F ( M 1 ) F ( M 2 ) F({M_1} \oplus {M_2}) = F({M_1})F({M_2}) and F ( M ) = a e F ( M ∖ e ) + b e F ( M / e ) F(M) = {a_e}F(M\backslash e) + {b_e}F(M/e) for e e not a loop or coloop of M M , where a e {a_e} , b e {b_e} are scalar parameters depending only on e e . We classify strong Tutte functions of all matroids into seven types, generalizing Brylawski’s classification of Tutte-Grothendieck invariants. One type is, like Tutte-Grothendieck invariants, an evaluation of a rank polynomial; all types are given by a Tutte polynomial. The classification remains valid if the domain is any minor-closed class of matroids containing all three-point matroids. Similar classifications hold for strong Tutte functions of colored matroids, where the parameters depend on the color of e e , and for strong Tutte functions of graphs and edge-colored graphs whose values do not depend on the attachments of loops. The latter classification implies new characterizations of Kauffman’s bracket polynomials of signed graphs and link diagrams.