Lower bound of one-ninth for plane-saturation ratios

Determine whether every planar graph with minimum degree 3 satisfies the lower bound \(\sat_P(G)/e(G)\geq 1/9\).

Background

The paper studies the plane-saturation number $\sat_P(G)$, defined as the minimum number of edges in a plane-saturated subgraph of a planar graph $G$. Earlier work established a lower bound of $e(G)/16$ for planar graphs without degree-1 or degree-2 twins, which in particular applies to maximal planar graphs.

The cited conjecture proposes improving this lower bound to a ratio of $1/9$ for all planar graphs of minimum degree 3. The paper constructs an infinite family of maximal planar graphs whose plane-saturation ratios approach $1/9$ from above, showing that the conjectured constant would be asymptotically sharp, but it does not establish the conjectured lower bound itself.