arXiv subjects

Sean McGuinness

Publications and source records attributed to Sean McGuinness.

9 recordsLinked to original sources

arXiv2026

Short Rainbow Circuits in Regular Matroids

DeVos et al conjectured that if $M$ is a simple, regular matroid and $c$ is a colouring of the elements of $M$ with $r(M)+1$ colours, where each colour class has at least two elements, then $M$ contains a rainbow circuit of size at most $\lceil \frac {r(M)+1}2 \rceil.$ We prove this conjecture by showing that for all such regular matroids there are four rainbow circuits $C_i,\ i = 1,2,3,4$ for which $\sum_i |C_i| \le 2r(M) +4$ and for which no element of $M$ belongs to more than two of the circuits.

math.CO↗

arXiv2024

Skew circuits and circumference in a binary matroid

Let C_1 and C_2 be skew circuits in a binary matroid having circumference c. For any positive integer k there is a constant a_k such that if min { |A| ; C_1 \subset A \subset E-A} > a_k, then |C_1| + |C_2| < 2c -k.

math.CO↗

arXiv2023

Cyclic Orderings of Paving Matroids

A matroid M is cyclically orderable if there is a cyclic permutation of the elements of M such that any r consecutive elements form a basis in M. An old conjecture of Kajitani, Miyano, and Ueno states that a matroid M is cyclically orderable if and only if for all nonempty subsets X in E(M), |X|/r(M) is less than or equal to |E(M)|/r(M). In this paper, we verify this conjecture for all paving matroids.

math.CO↗

arXiv2023

Serial Exchanges in Random Bases

It was conjectured by Kotlar and Ziv that for any two bases $B_1$ and $B_2$ in a matroid $M$ and any subset $X \subset B_1$, there is a subset $Y$ and orderings $x_1 \prec x_2 \prec \cdots \prec x_k$ and $y_1 \prec y_2 \prec \cdots \prec y_k$ of $X$ and $Y$, respectively, such that for $i = 1, \dots ,k$, $B_1 - \{ x_1, \dots ,x_i\} + \{y_1, \dots ,y_k \}$ and $B_2 - \{ y_1, \dots ,y_i\} + \{x_1, \dots ,x_k \}$ are bases; that is, $X$ is serially exchangeable with $Y$. Let $M$ be a rank-$n$ matroid which is representable over $\mathbb{F}_q.$ We show that for $q>2,$ if bases $B_1$ and $B_2$ are chosen randomly amongst all bases of $M$, and if a subset $X$ of size $k \le \ln(n)$ is chosen randomly in $B_1$, then with probability tending to one as $n \rightarrow \infty$, there exists a subset $Y\subset B_2$ such that $X$ is serially exchangeable with $Y.$

math.CO↗

arXiv2022

Rota's Basis Conjecture for Matroids with Density Close to One

Rota's basis conjecture (RBC) states that given a collection B of n bases in a matroid M of rank n, one can always find n disjoint rainbow bases with respect to B. We show that if M is a matroid having n + k elements, then one can construct n - k^3 disjoint rainbow bases, where b is a constant depending only on k.

math.CO↗

arXiv2020

Girth conditions and Rota's basis conjecture

Rota's basis conjecture (RBC) states that given a collection $\mathcal{B}$ of $n$ bases in a matroid $M$ of rank $n$, one can always find $n$ disjoint rainbow bases with respect to $\mathcal{B}$. In this paper, we show that if $M$ has girth at least $n-o(\sqrt{n})$, and no element of $M$ belongs to more than $o(\sqrt{n})$ bases in $\mathcal{B}$, then one can find at least $n - o(n)$ disjoint rainbow bases with respect to $\mathcal{B}$. This result can be seen as an extension of the work of Geelen and Humphries, who proved RBC in the case where $M$ is paving, and $\mathcal{B}$ is a pairwise disjoint collection. We make extensive use of the cascade idea introduced by Bucić et al.

math.CO↗

arXiv2020

Frame Matroids, Toric Ideals, and a Conjecture of White

Blasiak verified a conjecture of White for graphic matroids by showing that the toric ideal of a graphic matroid is generated by quadrics. In this paper, we extend this result to frame matroids satisfying a linearity condition. Such classes of matroids include graphic matroids, bicircular matroids, signed graphic matroids, and more generally frame matroids obtained from group-labelled graphs.

math.CO↗

arXiv2018

Isotope Harvesting at FRIB: Additional opportunities for scientific discovery

The Facility for Rare Isotope Beams (FRIB) at Michigan State University provides a unique opportunity to access some of the nation's most specialized scientific resources: radioisotopes. An excess of useful radioisotopes will be formed as FRIB fulfills its basic science mission of providing rare isotope beams. In order for the FRIB beams to reach high-purity, many of the isotopes are discarded and go unused. If harvested, the unused isotopes could enable cutting-edge research for diverse applications ranging from medical therapy and diagnosis to nuclear security. Given that FRIB will have the capability to create about 80 percent of all possible atomic nuclei, harvesting at FRIB will provide a fast path for access to a vast array of isotopes of interest in basic and applied science investigations. To fully realize this opportunity, infrastructure investment is required to enable harvesting and purification of otherwise unused isotopes. An investment in isotope harvesting at FRIB will provide the nation with a powerful resource for development of crucial isotope applications.

physics.ins-det↗

arXiv2016

A Dichotomy Theorem for Circular Colouring Reconfiguration

The "reconfiguration problem" for circular colourings asks, given two $(p,q)$-colourings $f$ and $g$ of a graph $G$, is it possible to transform $f$ into $g$ by changing the colour of one vertex at a time such that every intermediate mapping is a $(p,q)$-colouring? We show that this problem can be solved in polynomial time for $2\leq p/q <4$ and is PSPACE-complete for $p/q\geq 4$. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings.

math.CO↗