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Zach Walsh

Publications and source records attributed to Zach Walsh.

16 recordsLinked to original sources
arXiv2026

The unbreakable quasi-graphic matroids

A matroid M is unbreakable if it is connected and M/F is connected for every flat F of M . Oxley and Pfeil characterized the unbreakable graphic matroids, and Fife, Mayhew, Oxley, and Semple characterized the graphs underlying 3-connected unbreakable frame matroids. We extend the latter result by giving a complete characterization of the 3-connected unbreakable quasi-graphic matroids. As a special case we obtain a characterization of the 3-connected lifted-graphic matroids.

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arXiv2026

Matroids from gain graphs over quotient groups

We present a new construction for matroids from gain graphs that simultaneously generalizes several existing constructions. The construction takes as input a gain graph over a Frobenius group $\Gamma$ with Frobenius kernel $\Gamma_1$ and outputs an elementary lift of the frame matroid of the underlying gain graph over the quotient group $\Gamma/\Gamma_1$. While the hypothesis that $\Gamma$ is a Frobenius group may seem unusual, we prove that it is in some sense necessary: if $\Gamma$ is any finite group with a nontrivial proper normal subgroup $\Gamma_1$ and there is a construction that takes in a complete $\Gamma$-gain graph and outputs an elementary lift $M$ of the frame matroid of the underlying $(\Gamma/\Gamma_1)$-gain graph so that a cycle of the graph is a circuit of $M$ if and only if it is $\Gamma$-balanced, then $\Gamma$ is a Frobenius group with Frobenius kernel $\Gamma_1$.

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arXiv2025

Excluding a line from positroids

For all positive integers $\ell$ and $r$, we determine the maximum number of elements of a simple rank-$r$ positroid without the rank-$2$ uniform matroid $U_{2,\ell+2}$ as a minor, and characterize the matroids with the maximum number of elements. We prove this as a consequence of a more general result, which also determines the maximum number of elements of a simple rank-$r$ bicircular matroid, lattice path matroid, multi-path matroid, or colaminar matroid with no $U_{2,\ell+2}$-minor. This result continues a long line of research into upper bounds on the number of elements of matroids from various classes that forbid $U_{2,\ell+2}$ as a minor. This is the first paper to study positroids in this context, and it suggests methods to study similar problems for other classes of matroids, such as gammoids or base-orderable matroids.

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arXiv2025

The column number for 3-modular matrices

An integer-valued matrix $\mathbf{A}$ is $\Delta$-modular if each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix has determinant at most $\Delta$ in absolute value. The column number problem is to determine the maximum number of pairwise non-parallel columns of a rank-$r$, $\Delta$-modular matrix. Exact values for the column number are only known for $r \le 2$ or $\Delta \le 2$. We prove that if $r$ is sufficiently large, then the maximum number of pairwise non-parallel columns of a rank-$r$, $3$-modular matrix is $\binom{r+1}{2} + 2(r-1)$. This settles a conjecture by Lee, Paat, Stallknecht, and Xu on the column number in the case $\Delta = 3$. We complement this main result by showing that there are at least three $3$-modular matrices with pairwise non-isomorphic vector matroids that attain this upper bound. More generally, we show that if $r > \Delta$, then the number of $\Delta$-modular matrices with $\binom{r+1}{2} + (\Delta-1)(r-1)$ pairwise non-parallel columns and pairwise non-isomorphic vector matroids is at least exponential in $\sqrt{\Delta}$; previously only one matrix was known due to Lee et al.

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arXiv2025

Tur\'an densities for matroid basis hypergraphs

Let $U$ be a uniform matroid. For all positive integers $n$ and $r$ with $n \ge r$, what is the maximum number of bases of an $n$-element, rank-$r$ matroid without $U$ as a minor? We show that this question arises by restricting the problem of determining the Tur\'an number of a daisy hypergraph to the family of matroid basis hypergraphs. We then answer this question for several interesting choices of $U$.

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arXiv2023

Extended formulations for the integer hull of strictly $\Delta$-modular cographic polyhedral cones

Conforti et al. give a compact extended formulation for a class of bimodular-constrained integer programs, namely those that model the stable set polytope of a graph with no disjoint odd cycles. We extend their techniques to design compact extended formulations for the integer hull of translated polyhedral cones whose constraint matrix is strictly $\Delta$-modular and has rows that represent a cographic matroid. Our work generalizes the important special case from Conforti et al. concerning $4$-connected graphs with odd cycle transversal number at least $4$. We also discuss the necessity of our assumptions.

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arXiv2023

Dense circuit graphs and the planar Tur\'an number of a cycle

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_k$ denote the cycle of length $k$. The planar Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_k)$ is known for $k\le 7$. We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Tur\'an numbers. In particular, we prove that there is a constant $D$ so that $\textrm{ex}_{\mathcal P}(n,C_k) \le 3n - 6 - Dn/k^{\log_2^3}$ for all $k, n\ge 4$. When $k \ge 11$ this bound is tight up to the constant $D$ and proves a conjecture of Cranston, Lidick\'y, Liu, and Shantanam.

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arXiv2023

Planar Tur\'an number of the 7-cycle

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_{\ell}$ denote the cycle of length $\ell$. The planar Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_{\ell})$ behaves differently for $\ell\le 10$ and for $\ell\ge 11$, and it is known when $\ell \in \{3,4,5,6\}$. We prove that $\textrm{ex}_{\mathcal P}(n,C_7) \le \frac{18n}{7} - \frac{48}{7}$ for all $n > 38$, and show that equality holds for infinitely many integers $n$.

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arXiv2023

Matroid lifts and representability

A 1965 result of Crapo shows that every elementary lift of a matroid $M$ can be constructed from a linear class of circuits of $M$. In a recent paper, Walsh generalized this construction by defining a rank-$k$ lift of a matroid $M$ given a rank-$k$ matroid $N$ on the set of circuits of $M$, and conjectured that all matroid lifts can be obtained in this way. In this sequel paper we simplify Walsh's construction and show that this conjecture is true for representable matroids but is false in general. This gives a new way to certify that a particular matroid is non-representable, which we use to construct new classes of non-representable matroids. Walsh also applied the new matroid lift construction to gain graphs over the additive group of a non-prime finite field, generalizing a construction of Zaslavsky for these special groups. He conjectured that this construction is possible on three or more vertices only for the additive group of a non-prime finite field. We show that this conjecture holds for four or more vertices, but fails for exactly three.

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arXiv2022

On the column number and forbidden submatrices for $Δ$-modular matrices

An integer matrix $\mathbf{A}$ is $Δ$-modular if the determinant of each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix of $\mathbf{A}$ has absolute value at most $Δ$. The study of $Δ$-modular matrices appears in the theory of integer programming, where an open conjecture is whether integer programs defined by $Δ$-modular constraint matrices can be solved in polynomial time if $Δ$ is considered constant. The conjecture is only known to hold true when $Δ\in \{1,2\}$. In light of this conjecture, a natural question is to understand structural properties of $Δ$-modular matrices. We consider the column number question -- how many nonzero, pairwise non-parallel columns can a rank-$r$ $Δ$-modular matrix have? We prove that for each positive integer $Δ$ and sufficiently large integer $r$, every rank-$r$ $Δ$-modular matrix has at most $\binom{r+1}{2} + 80Δ^7 \cdot r$ nonzero, pairwise non-parallel columns, which is tight up to the term $80Δ^7$. This is the first upper bound of the form $\binom{r+1}{2} + f(Δ)\cdot r$ with $f$ a polynomial function. Underlying our results is a partial list of matrices that cannot exist in a $Δ$-modular matrix. We believe this partial list may be of independent interest in future studies of $Δ$-modular matrices.

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arXiv2022

Small cocircuits in minimally vertically $4$-connected matroids

Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when $k \le 3$. We show that every minimally vertically $4$-connected matroid with at least six elements has a $4$-element cocircuit, or a $5$-element cocircuit that contains a triangle, with the exception of a specific non-binary $9$-element matroid. Consequently, every minimally vertically $4$-connected binary matroid with at least six elements has a $4$-element cocircuit.

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arXiv2022

A new matroid lift construction and an application to group-labeled graphs

A well-known result of Brylawski constructs an elementary lift of a matroid $M$ from a linear class of circuits of $M$. We generalize this result by showing how to construct a rank-$k$ lift of $M$ from a rank-$k$ matroid on the set of circuits of $M$. We conjecture that every lift of $M$ arises via this construction. We then apply this result to group-labeled graphs, generalizing a construction of Zaslavsky. Given a graph $G$ with edges labeled by a group, Zaslavsky's lift matroid $K$ is an elementary lift of the graphic matroid $M(G)$ that respects the group-labeling; specifically, the cycles of $G$ that are circuits of $K$ coincide with the cycles that are balanced with respect to the group-labeling. For $k \ge 2$, when does there exist a rank-$k$ lift of $M(G)$ that respects the group-labeling in this same sense? For abelian groups, we show that such a matroid exists if and only if the group is isomorphic to the additive group of a non-prime finite field.

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arXiv2021

The extremal function for geometry minors of matroids over prime fields

A frame template over a field $\mathbb F$ describes the precise way in which a given $\mathbb F$-representable matroid is close to being a frame matroid. Our main result determines the maximum-rank projective or affine geometry that is described by a given frame template over a prime field. Subject to the matroid minors hypothesis of Geelen, Gerards, and Whittle, we use our result to determine, for each projective or affine geometry $N$ over a prime field $\mathbb F$, a best-possible upper bound on the number of elements in a simple $\mathbb F$-representable matroid $M$ of sufficiently large rank with no $N$-minor.

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arXiv2021

$2$-Modular Matrices

A rank-$r$ integer matrix $A$ is $\Delta$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $\Delta$. The class of $1$-modular, or unimodular, matrices is of fundamental significance in both integer programming theory and matroid theory. A 1957 result of Heller shows that the maximum number of nonzero, pairwise non-parallel rows of a rank-$r$ unimodular matrix is ${r + 1 \choose 2}$. We prove that, for each sufficiently large integer $r$, the maximum number of nonzero, pairwise non-parallel rows of a rank-$r$ $2$-modular matrix is ${r + 2 \choose 2} - 2$.

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arXiv2021

Excluding a line from $\mathbb C$-representable matroids

For each positive integer $t$ and each sufficiently large integer $r$, we show that the maximum number of elements of a simple, rank-$r$, $\mathbb C$-representable matroid with no $U_{2,t+3}$-minor is $t{r\choose 2}+r$. We derive this as a consequence of a much more general result concerning matroids on group-labeled graphs.

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