Largest 4-Connected Components of 3-Connected Planar Triangulations

Let Tn be a 3-connected n-vertex planar triangulation chosen uniformly at random. Then the number of vertices in the largest 4-connected component of Tn 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).

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