Arc diagrams, flip distances, and Hamiltonian triangulations [article]

Jean Cardinal and Michael Hoffmann and Vincent Kusters and Csaba D. Tóth and Manuel Wettstein
2016 arXiv   pre-print
We show that every triangulation (maximal planar graph) on n> 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
more » ... 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.
arXiv:1611.02541v2 fatcat:v2i76nkg4bejziekdmwh2bn6ru