Largest 4‐connected components of 3‐connected planar triangulations

Let T n be a 3‐connected n ‐vertex planar triangulation chosen uniformly at random. Then the number of vertices in the largest 4‐connected component of T n is asymptotic to n /2 with probability tending to 1 as n → ∞. It follows that almost all 3‐connected triangulations with n vertices have a cycle of length at least n /2 + o(n) .