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Register a new userThere are only a finite number of excluded minors for the class of bicircular matroids
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We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if $N$ is an excluded minor of rank at least eight, then $N$ is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that $N$ already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows.
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We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list $mathcal{E}$ of 18 such matroids, and show that if $N$ is such an excluded minor, then either $N in mathcal{E}$ or $N$ is a 2-sum of $U_{2,4}$ and a 3-connected non-binary frame matroid.
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let $mathcal{M}$ be one of these three classes, and let $r$ be a positive integer. We show that $mathcal{M}$ has only a finite number of excluded minors of rank $r$.
We give upper and lower bounds on the number of delta-matroids, and on the number of even delta-matroids.
The Hadwiger number $h(G)$ is the order of the largest complete minor in $G$. Does sufficient Hadwiger number imply a minor with additional properties? In [2], Geelen et al showed $h(G)geq (1+o(1))ctsqrt{ln t}$ implies $G$ has a bipartite subgraph with Hadwiger number at least $t$, for some explicit $csim 1.276dotsc$. We improve this to $h(G) geq (1+o(1))tsqrt{log_2 t}$, and provide a construction showing this is tight. We also derive improved bounds for the topological minor variant of this problem.
A partial Motzkin path is a path from $(0, 0)$ to $(n, k)$ in the $XOY$-plane that does not go below the $X$-axis and consists of up steps $U=(1, 1)$, down steps $D=(1, -1)$ and horizontal steps $H=(1, 0)$. A weighted partial Motzkin path is a partial Motzkin path with the weight assignment that all up steps and down steps are weighted by 1, the horizontal steps are endowed with a weight $x$ if they are lying on $X$-axis, and endowed with a weight $y$ if they are not lying on $X$-axis. Denote by $M_{n,k}(x, y)$ to be the weight function of all weighted partial Motzkin paths from $(0, 0)$ to $(n, k)$, and $mathcal{M}=(M_{n,k}(x,y))_{ngeq kgeq 0}$ to be the infinite lower triangular matrices. In this paper, we consider the sums of minors of second order of the matrix $mathcal{M}$, and obtain a lot of interesting determinant identities related to $mathcal{M}$, which are proved by bijections using weighted partial Motzkin paths. When the weight parameters $(x, y)$ are specialized, several new identities are obtained related to some classical sequences involving Catalan numbers. Besides, in the alternating cases we also give some new explicit formulas for Catalan numbers.
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