Oriented cobicircular matroids are $GSP$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910394237321216 |
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| author | Guzmán-Pro, Santiago Hochstättler, Winfried |
| author_facet | Guzmán-Pro, Santiago Hochstättler, Winfried |
| contents | Colourings and flows are well-known dual notions in Graph Theory. In turn, the definition of flows in graphs naturally extends to flows in oriented matroids. So, the colour-flow duality gives a generalization of Hadwiger's conjecture about graph colourings, to a conjecture about coflows of oriented matroids. The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-minor free oriented matroid, then $\mathcal{O}$ has a now-where $3$-coflow, i.e., it is $3$-colourable in the sense of Hochstättler-Nešetřil. The class of generalized series parallel ($GSP$) oriented matroids is a class of $3$-colourable oriented matroids with no $M(K_4)$-minor. So far, the only technique towards proving that all orientations of a class $\mathcal{C}$ of $M(K_4)$-minor free matroids are $GSP$ (and thus $3$-colourable), has been to show that every matroid in $\mathcal{C}$ has a positive coline. Towards proving Hadwiger's conjecture for the class of gammoids, Goddyn, Hochstättler, and Neudauer conjectured that every gammoid has a positive coline. In this work we disprove this conjecture by exhibiting an infinite class of strict gammoids that do not have positive colines. We conclude by proposing a simpler technique for showing that certain oriented matroids are $GSP$. In particular, we recover that oriented lattice path matroids are $GSP$, and we show that oriented cobicircular matroids are $GSP$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06591 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Oriented cobicircular matroids are $GSP$ Guzmán-Pro, Santiago Hochstättler, Winfried Combinatorics 05B35, 05C15 Colourings and flows are well-known dual notions in Graph Theory. In turn, the definition of flows in graphs naturally extends to flows in oriented matroids. So, the colour-flow duality gives a generalization of Hadwiger's conjecture about graph colourings, to a conjecture about coflows of oriented matroids. The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-minor free oriented matroid, then $\mathcal{O}$ has a now-where $3$-coflow, i.e., it is $3$-colourable in the sense of Hochstättler-Nešetřil. The class of generalized series parallel ($GSP$) oriented matroids is a class of $3$-colourable oriented matroids with no $M(K_4)$-minor. So far, the only technique towards proving that all orientations of a class $\mathcal{C}$ of $M(K_4)$-minor free matroids are $GSP$ (and thus $3$-colourable), has been to show that every matroid in $\mathcal{C}$ has a positive coline. Towards proving Hadwiger's conjecture for the class of gammoids, Goddyn, Hochstättler, and Neudauer conjectured that every gammoid has a positive coline. In this work we disprove this conjecture by exhibiting an infinite class of strict gammoids that do not have positive colines. We conclude by proposing a simpler technique for showing that certain oriented matroids are $GSP$. In particular, we recover that oriented lattice path matroids are $GSP$, and we show that oriented cobicircular matroids are $GSP$. |
| title | Oriented cobicircular matroids are $GSP$ |
| topic | Combinatorics 05B35, 05C15 |
| url | https://arxiv.org/abs/2209.06591 |