On the number of edges in saturated partial embeddings of maximal planar graphs
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arXiv
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| Format: | Preprint |
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2025
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| author | Barát, János Blázsik, Zoltán L. Keszegh, Balázs Zheng, Zeyu |
| author_facet | Barát, János Blázsik, Zoltán L. Keszegh, Balázs Zheng, Zeyu |
| contents | We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane subgraph of $G$ such that the addition of any edge either violates planarity or results in a graph that is not a subgraph of $G$. We focus on maximal planar graphs and establish an upper bound on $\mathrm{sat}_{\mathcal{P}}(G)$ by showing there exists a universal constant $ε> 0$ such that $\mathrm{sat}_{\mathcal{P}}(G) < (3-ε)v(G)$ for any maximal planar graph $G$ with $v(G) \geq 16$. This answers a question posed by Clifton and Simon. Additionally, we derive lower bound results and demonstrate that for maximal planar graphs with sufficiently large number of vertices, the minimum ratio $\mathrm{sat}_{\mathcal{P}}(G)/e(G)$ lies within the interval $(1/16, 1/9 + o(1)]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05438 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the number of edges in saturated partial embeddings of maximal planar graphs Barát, János Blázsik, Zoltán L. Keszegh, Balázs Zheng, Zeyu Combinatorics We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane subgraph of $G$ such that the addition of any edge either violates planarity or results in a graph that is not a subgraph of $G$. We focus on maximal planar graphs and establish an upper bound on $\mathrm{sat}_{\mathcal{P}}(G)$ by showing there exists a universal constant $ε> 0$ such that $\mathrm{sat}_{\mathcal{P}}(G) < (3-ε)v(G)$ for any maximal planar graph $G$ with $v(G) \geq 16$. This answers a question posed by Clifton and Simon. Additionally, we derive lower bound results and demonstrate that for maximal planar graphs with sufficiently large number of vertices, the minimum ratio $\mathrm{sat}_{\mathcal{P}}(G)/e(G)$ lies within the interval $(1/16, 1/9 + o(1)]$. |
| title | On the number of edges in saturated partial embeddings of maximal planar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.05438 |