arXiv subjects

James Dylan Douthitt

Publications and source records attributed to James Dylan Douthitt.

7 recordsLinked to original sources

arXiv2026

Weighted coloop splittings in rank six

With a view toward applications in Riemannian geometry, we explore coloop splitting properties of regular matroids. Nienhaus showed by classification in rank four that a regular matroid has a cocircuit whose deletion yields two coloops unless the matroid takes a particular form. In the latter case, one can split off any element of the ground set as a coloop. We reprove this using Seymour's structure theorem for regular matroids and prove an extension to matroids of ranks five and six. As an application to Riemannian geometry, we prove that the torus symmetry assumption in a recent result of Mouillé, Nienhaus, and the second author can be relaxed from rank ten to rank nine.

math.CO↗

arXiv2026

Degeneracy: From Graphs to Matroids

A graph is $k$-degenerate if every subgraph has a vertex of degree at most $k$. We extend this notion to matroids, defining a loopless matroid $M$ to be $k$-degenerate if every restriction of $M$ contains a cocircuit of size at most $k$; $M$ is minimally $k$-degenerate if it has cogirth $k$ and every proper restriction of $M$ has cogirth at most $k-1$. Our main result characterizes extremal minimally $k$-degenerate matroids. We also extend the known arboricity bound for matroids, showing that $k$-degenerate matroids have arboricity at most $k$ and providing sharper bounds.

math.CO↗

arXiv2026

Higher cosystoles of matroids

We define a matroid invariant called the three-cosystole that is related to higher notions of cogirth for weighted matroids, and we prove an optimal upper bound for it in the class of regular matroids of rank at most six. To accomplish this, we show that it is increasing under matroid extensions and then estimate it for each of the maximal simple regular matroids of rank at most six.

math.CO↗

arXiv2025

Rainbow triangles and the Erdős-Hajnal problem in projective geometries

We formulate a geometric version of the Erdős-Hajnal conjecture that applies to finite projective geometries rather than graphs, in both its usual 'induced' form and the multicoloured form. The multicoloured conjecture states, roughly, that a colouring $c$ of the points of $\mathsf{PG}(n-1,q)$ containing no copy of a fixed colouring $c_0$ of $\mathsf{PG}(k-1,q)$ for small $k$ must contain a subspace of dimension polynomial in $n$ that avoids some colour. If $(k,q) = (2,2)$, then $c_0$ is a colouring of a three-element 'triangle', and there are three essentially different cases, all of which we resolve. We derive both the cases where $c_0$ assigns the same colour to two different elements from a recent breakthrough result in additive combinatorics due to Kelley and Meka. We handle the case that $c_0$ is a 'rainbow' colouring by proving that rainbow-triangle-free colourings of projective geometries are exactly those that admit a certain decomposition into two-coloured pieces. This is closely analogous to a theorem of Gallai on rainbow-triangle-free coloured complete graphs. We also show that existing structure theorems resolve certain two-coloured cases where $(k,q) = (2,3)$, and $(k,q) = (3,2)$.

math.CO↗

arXiv2024

Classes of binary matroids with small lists of excluded induced minors

In earlier work, we characterized the class of matroids with no $M(C_4)$ as an induced minor and the class of matroids with no member of $\{M(C_4),M(K_4)\}$ as an induced minor. In this paper, for every two matroids in $\{M(C_4),M(K_4\backslash e),M(K_4),F_7\}$, we determine the class of matroids that have neither of the chosen pair as an induced minor. Additionally, we prove structural lemmas toward characterizing the class of matroids that do not contain $M(K_4)$ as an induced minor.

math.CO↗

arXiv2024

Chordal matroids arising from generalized parallel connections II

In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of $GF(q)$-chordal matroids as those matroids that can be constructed from projective geometries over $GF(q)$ by a sequence of generalized parallel connections across projective geometries over $GF(q)$. Our main result showed that when $q=2$, such matroids have no induced minor in $\{M(C_4),M(K_4)\}$. In this paper, we show that the class of $GF(2)$-chordal matroids coincides with the class of binary matroids that have none of $M(K_4)$, $M^*(K_{3,3})$, or $M(C_n)$ for $n\geq 4$ as a flat. We also show that $GF(q)$-chordal matroids can be characterized by an analogous result to Rose's 1970 characterization of chordal graphs as those that have a perfect elimination ordering of vertices.

math.CO↗

arXiv2023

Chordal matroids arising from generalized parallel connections

A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of simple $GF(q)$-representable matroids that can be built from projective geometries over $GF(q)$ by repeated generalized parallel connections across projective geometries. We show that this class of matroids is closed under induced minors. We characterize the class by its forbidden induced minors; the case when $q=2$ is distinctive.

math.CO↗