Zobrazeno 1 - 10
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pro vyhledávání: '"Walsh, Zach"'
We show that, under suitable hypotheses, the foundation of a generalized parallel connection of matroids is the relative tensor product of the foundations. Using this result, we show that the foundation of a 2-sum of matroids is the absolute tensor p
Externí odkaz:
http://arxiv.org/abs/2404.10656
We are motivated by integer linear programs (ILPs) defined by constraint matrices with bounded determinants. Such matrices generalize the notion of totally-unimodular matrices. When the determinants are bounded by $2$, the matrix is called bimodular.
Externí odkaz:
http://arxiv.org/abs/2311.06017
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{e
Externí odkaz:
http://arxiv.org/abs/2310.06631
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 $\
Externí odkaz:
http://arxiv.org/abs/2306.13594
Autor:
Bernstein, Daniel Irving, Walsh, Zach
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-$k$ m
Externí odkaz:
http://arxiv.org/abs/2306.12543
An integer matrix $\mathbf{A}$ is $\Delta$-modular if the determinant of each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix of $\mathbf{A}$ has absolute value at most $\Delta$. The study of $\Delta$-modular matrices appears in th
Externí odkaz:
http://arxiv.org/abs/2212.03819
Autor:
Oxley, James, Walsh, Zach
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 $k \le 3$.
Externí odkaz:
http://arxiv.org/abs/2110.13120
Autor:
Oxley, James, Walsh, Zach
Publikováno v:
SIAM J. Disc. Math. 36 (2022), 1231-1248
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 theo
Externí odkaz:
http://arxiv.org/abs/2105.04525
Autor:
Walsh, Zach
Publikováno v:
Electron. J. Combin. 29 (2022), Paper 1.6, 18pp
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 $M$
Externí odkaz:
http://arxiv.org/abs/2104.08257
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 consequenc
Externí odkaz:
http://arxiv.org/abs/2101.12000