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arXiv · 2502.05438

On the number of edges in saturated partial embeddings of maximal planar graphs

János Barát·Zoltán L. Blázsik·Balázs Keszegh·Zeyu Zheng
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Abstract

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 $\epsilon > 0$ such that $\mathrm{sat}_{\mathcal{P}}(G) < (3-\epsilon)v(G)$ 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 $(1/16, 1/9 + o(1)]$.

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János Barát, Zoltán L. Blázsik, Balázs Keszegh, Zeyu Zheng. 2025-02-08. On the number of edges in saturated partial embeddings of maximal planar graphs. https://arxiv.org/abs/2502.05438

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