Constructions stemming from non-separating planar graphs and their Colin de Verdière invariant
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arXiv
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| Natura: | Preprint |
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2021
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| _version_ | 1866913283784572928 |
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| author | Pavelescu, Andrei Pavelescu, Elena |
| author_facet | Pavelescu, Andrei Pavelescu, Elena |
| contents | A planar graph $G$ is said to be non-separating if there exists an embedding of $G$ in $\mathbb{R}^2$ such that for any cycle $\mathcal{C}\subset G$, all vertices of $G\setminus \mathcal{C}$ are within the same connected component of $\mathbb{R}^2\setminus \mathcal{C}$. Dehkordi and Farr classified the non-separating planar graphs as either outerplanar graphs, subgraphs of wheel graphs, or subgraphs of elongated triangular prisms. We use maximal non-separating planar graphs to construct examples of maximal linkless graphs and maximal knotless graphs. We show that for a maximal non-separating planar graph $G$ with $n\ge 7$ vertices, the complement $cG$ is $(n-7)-$apex. This implies that the Colin de Verdière invariant of the complement $cG$ satisfies $μ(cG) \le n-4$. We show this to be an equality. As a consequence, the conjecture of Kotlov, Lovàsz, and Vempala that for a simple graph $G$, $μ(G)+μ(cG)\ge n-2$ is true for 2-apex graphs $G$ for which $G-\{u,v\}$ is planar non-separating. It also follows that complements of non-separating planar graphs of order at least nine are intrinsically linked. We prove that the complements of non-separating planar graphs $G$ of order at least ten are intrinsically knotted. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_05740 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Constructions stemming from non-separating planar graphs and their Colin de Verdière invariant Pavelescu, Andrei Pavelescu, Elena Combinatorics Geometric Topology 57M15 (Primary), 05C10 (Secondary) A planar graph $G$ is said to be non-separating if there exists an embedding of $G$ in $\mathbb{R}^2$ such that for any cycle $\mathcal{C}\subset G$, all vertices of $G\setminus \mathcal{C}$ are within the same connected component of $\mathbb{R}^2\setminus \mathcal{C}$. Dehkordi and Farr classified the non-separating planar graphs as either outerplanar graphs, subgraphs of wheel graphs, or subgraphs of elongated triangular prisms. We use maximal non-separating planar graphs to construct examples of maximal linkless graphs and maximal knotless graphs. We show that for a maximal non-separating planar graph $G$ with $n\ge 7$ vertices, the complement $cG$ is $(n-7)-$apex. This implies that the Colin de Verdière invariant of the complement $cG$ satisfies $μ(cG) \le n-4$. We show this to be an equality. As a consequence, the conjecture of Kotlov, Lovàsz, and Vempala that for a simple graph $G$, $μ(G)+μ(cG)\ge n-2$ is true for 2-apex graphs $G$ for which $G-\{u,v\}$ is planar non-separating. It also follows that complements of non-separating planar graphs of order at least nine are intrinsically linked. We prove that the complements of non-separating planar graphs $G$ of order at least ten are intrinsically knotted. |
| title | Constructions stemming from non-separating planar graphs and their Colin de Verdière invariant |
| topic | Combinatorics Geometric Topology 57M15 (Primary), 05C10 (Secondary) |
| url | https://arxiv.org/abs/2101.05740 |