arXiv subjects
Publications and source records attributed to Jim Geelen.
The following notes are my attempt to clarify, to myself, the OpenAI proof of the Cycle Double Cover Conjecture. This write-up offers no new insights, but some people may find it easier to read.
We prove that any element in a matroid can be removed, by either deletion or contraction, in such a way that no tangle "splits".
Melchior's inequality implies that the average line-length in a simple, rank-$3$, real-representable matroid is less than $3$. A similar result holds for complex-representable matroids, using Hirzebruch's inequality, but with a weaker bound of $4$. We show that the average plane-size in a simple, rank-$4$, complex-representable matroid is bounded above by an absolute constant, unless the matroid is the direct-sum of two lines. We also prove that, for any integer $k$, in complex-representable matroids with rank at least $2k-1$, the average size of a rank-$k$ flat is bounded above by a constant depending only on $k$. Finally, we prove that, for any integer $r\ge 2$, the average flat-size in rank-$r$ complex-representable matroids is bounded above by a constant depending only on $r$. We obtain our results using a theorem, due to Ben Lund, that gives a good estimate on the number of rank-$k$ flats in a complex-representable matroid.
We show that, for any prime $p$ and integer $k \geq 2$, a simple GF($p$)-representable matroid with sufficiently high rank has a rank-$k$ flat which is either independent in $M$, or is a projective or affine geometry. As a corollary we obtain a Ramsey-type theorem for GF($p$)-representable matroids. For any prime $p$ and integer $k\ge 2$, if we $2$-colour the elements in any simple GF($p$)-representable matroid with sufficiently high rank, then there is a monochromatic flat with rank $k$.
The Sylvester-Gallai Theorem states that every rank-$3$ real-representable matroid has a two-point line. We prove that, for each $k\ge 2$, every complex-representable matroid with rank at least $4^{k-1}$ has a rank-$k$ flat with exactly $k$ points. For $k=2$, this is a well-known result due to Kelly, which we use in our proof. A similar result was proved earlier by Barak, Dvir, Wigderson, and Yehudayoff and later refined by Dvir, Saraf, and Wigderson, but we get slightly better bounds with a more elementary proof.
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.
We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field $\mathbb{F}_1$ and a field extension $\mathbb{F}_2$, if $\mathbb{F}_1$-representability is not equivalent to $\mathbb{F}_2$-representability, then each $\mathbb{F}_1$-representable matroid is a minor of a $\mathbb{F}_2$-representable excluded minor for $\mathbb{F}_1$-representability.
We prove that, for each circle graph $H$, every graph with sufficiently large rank-width contains a vertex-minor isomorphic to $H$.
We relate two conjectures that play a central role in the reported proof of Rota's Conjecture. Let $\mathbb F$ be a finite field. The first conjecture states that: the branch-width of any $\mathbb F$-representable $N$-fragile matroid is bounded by a function depending only upon $\mathbb F$ and $N$. The second conjecture states that: if a matroid $M_2$ is obtained from a matroid $M_1$ by relaxing a circuit-hyperplane and both $M_1$ and $M_2$ are $\mathbb F$-representable, then the branch-width of $M_1$ is bounded by a function depending only upon $\mathbb F$. Our main result is that the second conjecture implies the first.
We present a simpler proof of Naji's characterization of circle graphs.
We present infinite sequences of excluded minors for both the class of lifted-graphic matroids and the class of frame matroids.
We prove that, if $B_1, \dots, B_n$ are disjoint bases of a rank-$n$ matroid, then there are at least $\lfloor{\frac{n}{6 \lceil{\log n}\rceil}}\rfloor$ disjoint transversals of $(B_1, \dots, B_n)$ that are also bases.
Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, {\em quasi-graphic matroids}, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every $3$-connected representable quasi-graphic matroid is either a lifted-graphic matroid or a frame matroid.
A graph has tree-width at most $k$ if it can be obtained from a set of graphs each with at most $k+1$ vertices by a sequence of clique sums. We refine this definition by, for each non-negative integer $θ$, defining the $θ$-tree-width of a graph to be at most $k$ if it can be obtained from a set of graphs each with at most $k+1$ vertices by a sequence of clique sums on cliques of size less than $θ$. We find the unavoidable minors for the graphs with large $θ$-tree-width and we obtain Robertson and Seymour's Grid Theorem as a corollary.
Let $Σ$ be a surface with boundary $b(Σ)$, $\mathcal{L}$ be a collection of $k$ disjoint $b(Σ)$-paths in $Σ$, and $P$ be a non-separating $b(Σ)$-path in $Σ$. We prove that there is a homeomorphism $ϕ: Σ\to Σ$ that fixes each point of $b(Σ)$ and such that $ϕ(\mathcal{L})$ meets $P$ at most $2k$ times. With this theorem, we derive explicit constants in the graph minor algorithms of Robertson and Seymour. We reprove a result concerning redundant vertices for graphs on surfaces, but with explicit bounds. That is, we prove that there exists a computable integer $t:=t(Σ,k)$ such that if $v$ is a '$t$-protected' vertex in a surface $Σ$, then $v$ is redundant with respect to any $k$-linkage.
We show that, for each real number $ε> 0$ there is an integer $c$ such that, if $M$ is a simple triangle-free binary matroid with $|M| \ge (\tfrac{1}{4} + ε) 2^{r(M)}$, then $M$ has critical number at most $c$. We also give a construction showing that no such result holds for any real number less than $\tfrac{1}{4}$. This shows that the "critical threshold" for the triangle is $\tfrac 1 4$. We extend the notion of critical threshold to every simple binary matroid $N$ and conjecture that, if $N$ has critical number $c\ge 3$, then $N$ has critical threshold $1-i\cdot 2^{-c}$ for some $i\in \{2,3,4\}$. We give some support for the conjecture by establishing lower bounds.
The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$. The Growth-Rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential, or infinite. Morover, if the growth-rate function is quadratic, then $h(r)\ge \binom{r+1}{2}$, with the lower bound coming from the fact that such classes necessarily contain all graphic matroids. We characterise the classes that satisfy $h(r) = \binom{r+1}{2}$ for all sufficiently large $r$.
Let $s,n \ge 2$ be integers. We give a qualitative structural description of every matroid $M$ that is spanned by a frame matroid of a complete graph and has no $U_{s,2s}$-minor and no rank-$n$ projective geometry minor, showing that every such matroid is `close' to a frame matroid. We also give a similar description of every matroid $M$ with a spanning projective geometry over a field GF$(q)$ as a restriction and with no $U_{s,2s}$-minor and no PG$(n,q')$-minor for any $q' > q$, showing that such an $M$ is `close' to a GF$(q)$-representable matroid.