An infinite family of excluded minors for strong base-orderability
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arXiv
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| Format: | Preprint |
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2015
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| _version_ | 1866911977642655744 |
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| author | Bonin, Joseph E. Savitsky, Thomas J. |
| author_facet | Bonin, Joseph E. Savitsky, Thomas J. |
| contents | We discuss a conjecture of Ingleton on excluded minors for base-orderability, and, extending a result he stated, we prove that infinitely many of the matroids that he identified are excluded minors for base-orderability, as well as for the class of gammoids. We prove that a paving matroid is base-orderable if and only if it has no minor that is isomorphic to the cycle matroid of the complete graph on four vertices. For each k that is at least 2, we define the property of k-base-orderability, which lies strictly between base-orderability and strong base-orderability, and we show that k-base-orderable matroids form what Ingleton called a complete class. By generalizing an example of Ingleton, we construct a set of matroids, each of which is an excluded minor for k-base-orderability, but is (k-1)-base-orderable; the union of these sets, over all k, is an infinite set of base-orderable excluded minors for strong base-orderability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1507_05521 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | An infinite family of excluded minors for strong base-orderability Bonin, Joseph E. Savitsky, Thomas J. Combinatorics 05B35 We discuss a conjecture of Ingleton on excluded minors for base-orderability, and, extending a result he stated, we prove that infinitely many of the matroids that he identified are excluded minors for base-orderability, as well as for the class of gammoids. We prove that a paving matroid is base-orderable if and only if it has no minor that is isomorphic to the cycle matroid of the complete graph on four vertices. For each k that is at least 2, we define the property of k-base-orderability, which lies strictly between base-orderability and strong base-orderability, and we show that k-base-orderable matroids form what Ingleton called a complete class. By generalizing an example of Ingleton, we construct a set of matroids, each of which is an excluded minor for k-base-orderability, but is (k-1)-base-orderable; the union of these sets, over all k, is an infinite set of base-orderable excluded minors for strong base-orderability. |
| title | An infinite family of excluded minors for strong base-orderability |
| topic | Combinatorics 05B35 |
| url | https://arxiv.org/abs/1507.05521 |