Loopless Matroid Research Articles

Adjoints of matroids

We first show that an adjoint of a loopless matroid is connected if and only if the original matroid is connected. By proving that the opposite lattice of a modular matroid is isomorphic to its extension lattice, we obtain that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice. This makes projective geometries become a key ingredient in characterizing the adjoint sequence ad0M,adM,ad2M,… of a connected matroid M. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to infinite projective geometries.

The tropical Poincaré-Hopf theorem

We express the beta invariant of a loopless matroid as tropical self-intersection number of the diagonal of its matroid fan (a “local” Poincaré-Hopf theorem). This provides another example of uncovering the “geometry” of matroids by expressing their invariants in terms of tropicalised geometric constructions. We also prove a global Poincaré-Hopf theorem and initiate the study of a more general tropical Lefschetz-Hopf trace formula by proving the two special cases of tropical curves and tropical tori.

The Free m-Cone of a Matroid and Its $${\mathcal {G}}$$-Invariant

For a matroid, its configuration determines its \({\mathcal {G}}\)-invariant. Few examples are known of pairs of matroids with the same \({\mathcal {G}}\)-invariant but different configurations. In order to produce new examples, we introduce the free m-cone \(Q_m(M)\) of a loopless matroid M, where m is a positive integer. We show that the \({\mathcal {G}}\)-invariant of M determines the \({\mathcal {G}}\)-invariant of \(Q_m(M)\), and that the configuration of \(Q_m(M)\) determines M; so if M and N are nonisomorphic and have the same \({\mathcal {G}}\)-invariant, then \(Q_m(M)\) and \(Q_m(N)\) have the same \({\mathcal {G}}\)-invariant but different configurations. We prove analogous results for several variants of the free m-cone. We also define a new matroid invariant of M, and show that it determines the Tutte polynomial of \(Q_m(M)\).

On Density-Critical Matroids

For a matroid $M$ having $m$ rank-one flats, the density $d(M)$ is $\tfrac{m}{r(M)}$ unless $m = 0$, in which case $d(M)= 0$. A matroid is density-critical if all of its proper minors of non-zero rank have lower density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among simple matroids that cannot be covered by $k$ independent sets is density-critical. It is straightforward to show that $U_{1,k+1}$ is the only minor-minimal loopless matroid with no covering by $k$ independent sets. We prove that there are exactly ten minor-minimal simple obstructions to a matroid being able to be covered by two independent sets. These ten matroids are precisely the density-critical matroids $M$ such that $d(M) > 2$ but $d(N) \le 2$ for all proper minors $N$ of $M$. All density-critical matroids of density less than $2$ are series-parallel networks. For $k \ge 2$, although finding all density-critical matroids of density at most $k$ does not seem straightforward, we do solve this problem for $k=\tfrac{9}{4}$.

Extensions of matroid covering and packing

Let M be a loopless matroid on E with rank function rM. Let β(M)=max0̸≠X⊆E|X|rM(X) and φ(M)=minrM(X)<r(M)|E|−|X|rM(E)−rM(X). The Matroid Covering and Packing Theorems state that the minimum number of independent sets whose union is E is ⌈β(M)⌉, and the maximum number of disjoint bases is ⌊φ(M)⌋.We prove refinements of these theorems. If β(M)=k+ε, where k≥0 is an integer and 0≤ε<1, then E can be partitioned into k+1 independent sets with one of size at most ε⋅rM(E). If φ(M)=k+ε, then M has k+1 disjoint independent sets such that k are bases and the other has size at least ε⋅rM(E).We apply these results to truncations of cycle matroids to refine graph-theoretic results due to Chen, Koh, and Peng in 1993 and to Chen and Lai in 1996.

The coloring game on matroids

A coloring of the ground set of a matroid is proper if elements of the same color form an independent set. For a loopless matroid M, its chromatic numberχ(M) is the minimum number of colors in a proper coloring. In this note we study a game-theoretic variant of this parameter.Suppose that Alice and Bob alternately properly color the ground set of a matroid M using a fixed set of colors. The game ends when the whole matroid has been colored, or if they arrive at a partial coloring that cannot be further properly extended. Alice wins in the first case, while Bob in the second. The game chromatic number of M, denoted by χg(M), is the minimum size of the set of colors for which Alice has a winning strategy. Clearly, χg(M)≥χ(M).We prove an upper bound χg(M)≤2χ(M) for every matroid M. This improves and extends a result of Bartnicki, Grytczuk and Kierstead [1], who showed that χg(M)≤3χ(M) holds for graphic matroids. Our bound is almost tight, as we construct a family of matroids Mk (for k≥3) satisfying χ(Mk)=k and χg(Mk)=2k−1, which improves a construction of Bartnicki et al. by 1.

The minimum number of minimal codewords in an [formula omitted]-code and in graphic codes

We survey some lower bounds on the function in the title based on matroid theory and address the following problem by Dosa et al. (2004): Determine the smallest number of circuits in a loopless matroid with no parallel elements and with a given size and rank. In the graphic 3-connected case we provide a lower bound which is a product of a linear function of the number of vertices and an exponential function of the average degree. We also prove that, for p≥38, every 3-connected graph with p vertices has at least as many cycles as the wheel with p vertices.

Indicated coloring of matroids

A coloring of a matroid is proper if elements of the same color form an independent set. For a loopless matroid M, its chromatic number χ(M) is the minimum number of colors in a proper coloring of M. In this note we study a game-theoretic variant of this parameter proposed by Grytczuk. In each round of the game Alice indicates an uncolored yet element, then Bob colors it properly using a color from a fixed set of colors C. The game ends if the whole matroid is colored or if Bob arrives at a partial coloring that cannot be properly extended. Alice wins in the first case, while Bob in the second. The indicated chromatic number of M, denoted by χi(M), is the minimum size of the set of colors C for which Alice has a winning strategy. We prove that for every loopless matroid M there is an equality χi(M)=χ(M).

Recurrence relations for the spectrum polynomial of a matroid

Combinatorial Laplace operators on the simplicial complex of independent sets of a loopless matroid M are known to have non-negative integral spectra (J. Amer. Math. Soc. B 13 (2000) 129). The spectrum polynomial of M, a polynomial in two variables formulated via the flats of M, is a generating function for the spectra of these operators. In this paper, we establish recurrence formulas for the spectrum polynomial of a matroid, analogous to the deletion–contraction recursions for the Tutte polynomial. However, we show that for any matroid M and e∈ M the new formulas for the spectrum polynomial depend on whether or not e is closed in M. In particular, the spectrum polynomial is a new invariant for matroids that is not a Tutte–Grothendieck invariant.

The Zero-Free Intervals for Characteristic Polynomials of Matroids

Let M be a loopless matroid with rank r and c components. Let P(M, t) be the characteristic polynomial of M. We shall show that (−1)rP(M, t)[ges ](1−t)r for t∈(−∞, 1), that the multiplicity of the zeros of P(M, t) at t=1 is equal to c, and that (−1)r+cP(M, t)[ges ](t−1)r for t∈(1, 32/27]. Using a result of C. Thomassen we deduce that the maximal zero-free intervals for characteristic polynomials of loopless matroids are precisely (−∞, 1) and (1, 32/27].

On matroid connectivity

If M is a loopless matroid in which M/ vbX and M/ vbY are connected and X∩ Y is non-empty, then one easily shows that M/ vb( X∪ Y) is connected. Likewise, it is straightforward to show that if G and H are n- connected graphs having at least n common vertices, then G ∪ H is n-connected. The purpose of this note is to prove a matroid connectivity result that is a common generalization of these two observations.

A Constructive Approach to the Critical Problem for Matroids

Let M be a loopless matroid on a finite non-empty set I which is representable over a finite field F . Let A be a representation of M over F , i.e. A is a family ( A i , i ∈ I ) indexed by I of vectors of F n (for some n ⩾ 1) such that J ⊆ I is an independent set of M iff ( A i , i ∈ J ) is a linearly independent family of vectors of F n . The critical number of A, c (A) , is the smallest integer k ⩾ 1 such that there exist k linear functionals f j : F n → F ( j ∈ {1, ... , k }) with the following property: ∀ i ∈ I , ∃ j ∈ {1, ... , k } such that f j ( A i ) ≠ 0. It is known that all representations of M over F have the same critical number: by definition this number is the critical number of M over F and is denoted by c F ( M ). The critical problem (determine c F ( M ) for a given finite field F and for a given class of matroids M representable over F ) contains as special cases some of the most fascinating problems in graph theory. We present here, for any k ⩾ 2 and finite field F , a simple constructive characterization of matroids M representable over F with c F ( M ) ⩾ k . Our result is quite similar to the Theorem of Hajós which gives a constructive characterization of simple graphs with chromatic number at least q (∀ q ⩾3).

On Hartfiel and Maxson's definition of connected sets of a matroid

In a loopless matroid a non-empty set is connected in the sense of Hartfiel and Maxson if and only if it spans a non-separable flat.

Cocircuit coverings and packings for binary matroids

If M is an arbitrary loopless matroid with ground set E(M) and rank function ρ, then let α(M) be the minimum size of a set of cocircuits of M, whose union is E(M), and let β(M) be the maximum size of a set of pairwise disjoint cocircuits of M. The following conjecture is based on Gallai's theorem that the vertex-stability and vertex-covering numbers of a graph G sum to the number of vertices of G (1).