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

TL;DR

This paper studies the minimal number of edges in saturated partial embeddings of maximal planar graphs, establishing bounds that answer a previously posed question and providing insights into their extremal properties.

Contribution

It provides an upper bound on the plane-saturation number for maximal planar graphs and derives bounds on the ratio of saturation edges to total edges, addressing open questions in the field.

Findings

Existence of a universal constant epsilon such that saturation number is less than (3 - epsilon) times vertices.

Lower bounds on the saturation ratio for large maximal planar graphs.

The ratio of saturation edges to total edges lies within (1/16, 1/9 + o(1)] for large graphs.

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 $(1/16, 1/9 + o(1)]$.