On the Number of Drawings of a Combinatorial Triangulation

Abstract In 1962, Tutte provided a formula for the number of combinatorial triangulations, that is, maximal planar graphs with a fixed triangular face and n additional vertices. In this note, we study in how many ways a combinatorial triangulation can be drawn as geometric triangulation, that is, with straight-line segments, on a given point set in the plane. Our central contribution is that there exist a combinatorial triangulation T and a point set S such that T can be drawn on S in at least $$\varOmega (1,31^n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>31</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ways as different geometric triangulations. We also show an upper bound on the number of drawings of a combinatorial triangulation on the so-called double chain point set.