On edge-disjoint empty triangles of point sets

Let P be a set of points in the plane in general position. Any three points x, y, x 2 P determine a triangle (x, y, z) of the plane. We say that (x, y, z) is empty if its interior contains no element of P . In this paper we study the following problems: What is the size of the largest family of edge-disjoint triangles of a point set? How many triangulations of P are needed to cover all the empty triangles of P? What is the largest number of edge-disjoint triangles of P containing a point q of the plane in their interior? Introduction Let P be a set of n points on the plane in general position. A geometric graph on P is a graph G whose vertices are the elements of P , two of which are adjacent if they are joined by a straight line segment. We say that G is plane if it has no edges that cross each other. A triangle of G consists of three points x, y, z 2 P such that xy, yz, and zx are edges of G; we will denote it as (x, y, z). If in addition (x, y, z) contains no elements of P in its interior, we say that it is empty. In a similar way, we say that, if x, y, z 2 P , then (x, y, z) is a triangle of P , and that xy, yz, and zx are the edges of (x, y, z). If (x, y, z) is empty, it is called a 3-hole of P . A 3-hole of P can be thought of as an empty triangle of the complete geometric graph KP on P . We remark that (x, y, z) will denote a triangle of a geometric graph, and also a triangle of a point set. A well-known result in graph theory says that, for n = 6k + 1 or n = 6k + 3, the edges of the complete graph Kn on n vertices can be decomposed into a set of n 2 /3 edge-disjoint triangles. These decompositions are known as Steiner triple systems [18]; see also Kirkman’s schoolgirl problem [12, 17]. In this paper, we address some variants of that problem, but for geometric graphs. Given a point set P , let (P ) be the size of the largest set of edge-disjoint empty triangles of P . It is clear that, if P is in convex position and it has n = 6k + 1 or n = 6k + 3 elements, then (P ) = n 2 /3. On the other hand, we prove that, for some point sets, namely Horton point sets, (P ) is O(n log n). We then study the problem of covering the empty triangles of point sets with as few triangulations of P as possible. For point sets in convex position, we prove that we need essentially n 3 /4 triangulations; our bound is tight. We also show that there are point 1Partially supported by project SEP-CONACYT of Mexico, Proyecto 80268. 3Partially supported by projects MTM2006-03909 (Spain) and SEP-CONACYT 80268 (Mexico). CRM Documents, vol. 8, Centre de Recerca Matemàtica, Bellaterra (Barcelona), 2011 15 16 Empty edge-disjoint triangles sets P for which O(n log n) triangulations are sufficient to cover all the empty triangles of P for a given point set P . Finally, we consider the problem of finding a point contained in the interior of many edge-disjoint triangles of P . We prove that for any point set there is a point contained in at least n2/12 edge-disjoint triangles. Furthermore, any point in the plane is contained in at most n2/9 edge-disjoint triangles of P , and this bound is sharp. In particular, we show that this bound is attained when P is the set of vertices of a regular polygon. Preliminary work The study of counting and finding k-holes in point sets has been an active area of research since Erdős and Szekeres [6, 7] asked about the existence of k-holes in planar point sets. It is known that any point set with at least ten points contains 5-holes; e.g. see [9]. Horton [10] proved that for k 7 there are point sets containing no k-holes. The question of the existence of 6-holes remained open for many years, but recently Nicolás [14] proved that any point set with sufficiently many points contains a 6-hole. A second proof of this result was subsequently given by Gerken [8]. The study of properties of the set of triangles generated by point sets on the plane has been of interest for many years. Let fk(n) be the minimum number of k-holes that a point set has. Clearly a point set has a minimum of f 3 (n) empty triangles. Katchalski and Meir [11] proved that n