Contractible Edges in \(4\)-Connected Maximal Planar Graphs

The fact that any \(n\)-vertex \(4\)-connected maximal planar graph admits at least \(\frac{3n+6}{5}\) \(4\)-contractible edges readily follows from the general results of W.D. McCuaig [9], [10] ,[11] and of L. Andersen, H. Fleischner, and B. Jackson [1].

Here we prove a lower bound of \(\lceil\frac{3n}{4}\rceil\) on the number of \(4\)-contractible edges in every \(4\)-connected maximal planar graph with at least eight vertices.