Arc diagrams, flip distances, and Hamiltonian triangulations

We show that every triangulation (maximal planar graph) on $n\ge 6$ vertices can be flipped into a Hamiltonian triangulation using a sequence of less than $n/2$ combinatorial edge flips. The previously best upper bound uses $4$-connectivity as a means to establish Hamiltonicity. But in general about $3n/5$ flips are necessary to reach a $4$-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on $n$ vertices from $5.2n-33.6$ to $5n-23$. We also show that for every triangulation on $n$ vertices there is a simultaneous flip of less than $2n/3$ edges to a $4$-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on $n$ vertices admits an arc diagram with less than $n/2$ biarcs, that is, after subdividing less than $n/2$ (of potentially $3n-6$) edges the resulting graph admits a $2$-page book embedding.

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