# Quadrangulations on 3-Colored Point Sets with Steiner Points and Their Winding Numbers

## Abstract

Let *P* be a point set on the plane, and consider whether *P* is *quadrangulatable*, that is, whether there exists a 2-connected plane graph *G* with each edge a straight segment such that *V*(*G*) = *P*, that the outer cycle of *G* coincides with the convex hull Conv(*P*) of *P*, and that each finite face of *G* is quadrilateral. It is easy to see that it is possible if and only if an even number of points of *P* lie on Conv(*P*). Hence we give a *k*-coloring to *P*, and consider the same problem, avoiding edges joining two vertices of *P* with the same color. In this case, we always assume that the number of points of *P* lying on Conv(*P*) is even and that any two consecutive points on Conv(*P*) have distinct colors. However, for every *k* ≥ 2, there is a *k*-colored non-quadrangulatable point set *P*. So we introduce *Steiner points*, which can be put in any position of the interior of Conv(*P*) and each of which may be colored by any of the *k* colors. When *k* = 2, Alvarez et al. proved that if a point set *P* on the plane consists of \({\frac{n}{2}}\) red and \({\frac{n}{2}}\) blue points in general position, then adding Steiner points *Q* with \({|Q| \leq \lfloor \frac{n-2}{6} \rfloor + \lfloor \frac{n}{4} \rfloor +1}\) , *P* ∪ *Q* is quadrangulatable, but there exists a non-quadrangulatable 3-colored point set for which no matter how many Steiner points are added. In this paper, we define the *winding number* for a 3-colored point set *P*, and prove that a 3-colored point set *P* in general position with a finite set *Q* of Steiner points added is quadrangulatable if and only if the winding number of *P* is zero. When *P* ∪ *Q* is quadrangulatable, we prove \({|Q| \leq \frac{7n+34m-48}{18}}\) , where |*P*| = *n* and the number of points of *P* in Conv(*P*) is 2*m*.

### Keywords

Quadrangulation Plane Point set 3-Coloring Steiner point Winding number## Preview

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