Related papers: Unavoidable Minors of Matroids with Minimum Cocirc…

For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from the complete graph $K_t$ by removing $s$ edges. A graph $G$ has no $\mathcal{K}_t^{-s}$ minor if it has no $H$ minor for every $H\in…

We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular…

In this sequel to "Foundations of matroids - Part 1", we establish several presentations of the foundation of a matroid in terms of small building blocks. For example, we show that the foundation of a matroid M is the colimit of the…

For each odd integer $k\ge 5$, we prove that, if $M$ is a simple rank-$r$ binary matroid with no odd circuit of length less than $k$ and with $|M| > k 2^{r-k+1}$, then $M$ is isomorphic to a restriction of the rank-$r$ binary affine…

Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if a binary matroid of rank $r$ is colored with exactly $r$ colors,…

We prove that the complement of any non-separating planar graph of order $2n-3$ contains a $K_n$ minor, and argue that the order $2n-3$ is lowest possible with this property. To illustrate the necessity of the non-separating hypothesis, we…

Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent…

Antimatroids were discovered by Dilworth in the context of lattices [4] and introduced by Edelman and Jamison as convex geometries in[5]. The author of the current paper independently discovered (possibly infinite) antimatroids in the…

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…

We prove that the non-regular binary matroids with no $P_9^*$-minor have linear growth rate and the maximum size binary matroids with no $P_9^*$-minor are graphic. The main technique in the proof is the Strong Splitter Theorem using which…

If $\mathcal{C}$ is a minor-closed class of matroids, then the class $\widetilde{\mathcal{C}}'_k$ of $k$-polymatroids whose $k$-natural matroids are in $\mathcal{C}$ is also minor-closed. We investigate the following question: When…

A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of…

Brylawski and Seymour independently proved that if $M$ is a connected matroid with a connected minor $N$, and $e \in E(M) - E(N)$, then $M \backslash e$ or $M / e$ is connected having $N$ as a minor. This paper proves an analogous but…

Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite…

We provide a formula for the Ehrhart polynomial of the connected matroid of size $n$ and rank $k$ with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $h^*$-real-rooted (and…

Thomassen, in 1983, conjectured that for a positive integer $k$, every $2$-connected non-bipartite graph of minimum degree at least $k + 1$ contains cycles of all lengths modulo $k$. In this paper, we settle this conjecture affirmatively.

We consider the situation that M and N are 3-connected matroids such that |E(N)| > 3 and C* is a cocircuit of M with the property that M/y has an N-minor for some y in C*. We show that either there is an element x in C* such that si(M/x) or…

We show that a simple rank-$r$ matroid with no $(t+1)$-element independent flat has at least as many elements as the matroid $M_{r,t}$ defined as the direct sum of $t$ binary projective geometries whose ranks pairwise differ by at most $1$.…

We consider the rank reduction problem for matroids: Given a matroid M and an integer k, find a minimum size subset of elements of M whose removal reduces the rank of M by at least k. When M is a graphical matroid this problem is the…

The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-free oriented matroid, then $\mathcal{O}$ admits a NZ $3$-coflow, i.e., it is $3$-colourable in the sense of…