@MISC{Alvarez12parity-constrainedtriangulations, author = {Victor Alvarez}, title = {Parity-constrained Triangulations with Steiner points}, year = {2012} }

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Abstract

Let P ⊂ R 2 be a set of n points, of which k lie in the interior of the convex hull CH(P) of P. Let us call a triangulation T of P even (odd) if and only if all its vertices have even (odd) degree, and pseudo-even (pseudo-odd) if at least the k interior vertices have even (odd) degree. On the one hand, triangulations having all its interior vertices of even degree have one nice property; their vertices can be 3-colored,see [1, 2, 3]. On the other hand, odd triangulationshave recently found an application in the colored version of the classic “Happy Ending Problem ” of Erdős and Szekeres, see [4]. In this paper we show that there are sets of points that admit neither pseudoeven nor pseudo-odd triangulations. Nevertheless, we show how to construct a set of SteinerpointsS = S(P)ofsizeatmost k 3 +c,wherecisapositiveconstant,suchthat a pseudo-even (pseudo-odd) triangulation can be constructed on P ∪S. Moreover, we alsoshow that even (odd) triangulationscan alwaysbe constructed using at most n 3 +c Steiner points, where again c is a positive constant. Our constructions have the property that every Steiner point lies in the interior of CH(P). 1