On two lower bound constructions

We address two problems. In the first, we refine the analysis of a lower bound construction of a point set having many non-crossing spanning trees. We also give more precise upper bounds on the maximum number of non-crossing spanning trees, perfect matchings and simple polygons of a planar point set. In the second, we give an improved lower bound construction for the d-interval problem. 1 Non-crossing subgraphs – an introduction Consider a set of n points in the plane and the straightline drawing of the complete graph Kn they define. A subgraph ofKn in this drawing is called non-crossing (or crossing-free) if its edges intersect only at common vertices. Ajtai, Chvátal, Newborn and Szemerédi proved that the number of non-crossing subgraphs of any drawing of Kn (even without the rectilinear restriction on edges) is bounded from above by 10. This result was a consequence of a lower bound on the crossing number of a graph G (i.e. the minimum number of crossing pairs of edges over all planar drawings of G). An improvement in this lower bound on crossing number [PT97] lead to an improved upper bound of 53000 on the number of non-crossing subgraphs. Further improvements on the number of non-crossing subgraphs have been obtained by bounding the number of triangulations of a planar point set. As noted in [GNT95], since every non-crossing subgraph can be extended to a triangulation, and since a triangulation has at most 3n edges, a bound of α on the number of triangulations implies a bound of 2α = (8α) on the number of non-crossing subgraphs. Smith [S89] proved a bound of 173000 on the number of triangulations of any planar point set with n points. An improved bound (on the number of triangulations) of 212.245113n−Θ(logn) ≤ 4855 was found by Seidel [S99], which was further improved by Denny and Sohler [DS97]. They proved a bound of 2 ≤ 279. On the other hand, lower bounds on the maximum number of non-crossing subgraphs are provided by specific configurations of points (see [GNT95] for a review of results and for the latest improvements at this time, that we are aware of). First [A79], then Hayward [H87], later Garćıa and Tejel [GT99] gave lower bounds of 2.27, 3.26 and 3.605 respectively on the number of simple polygons. The best bounds we know are from [GNT95]: 4.642 for simple polygons, Ω(8/n) = Ω((8− )) for triangulations, Ω(3/n) = Ω((3− )) for perfect matchings and Ω(9.35) for spanning trees ( > 0 is arbitrarily small). Here we improve the analysis of a construction given in [GNT95], obtaining a better lower bound on the number of non-crossing spanning trees. We also get more precise upper bounds for these three types of subgraphs (polygons, matchings and trees) as derived from an upper bound on the number of triangulations. 2 A lower bound for spanning trees We give a sharper analysis of a configuration S of n points given in [GNT95]. The parameters α, β, γ are to be specified later. The points of S are partitioned into two convex chains with opposite concavity (see Figure 1), |S1| = αn points on the upper chain C1 and |S2| = (1 − α)n points on the lower chain C2. Take any partition S1 = T1 ∪ F1 of the points in C1 with |T1| = (α− β)n, |F1| = βn. Select any subset M1 from T1 with |M1| = γn. The points (vertices) in T1, F1,M1 are called tree points, free points and matched points, respectively. Matched points are labeled by “m”, (the rest of tree vertices are unlabeled) and the free points are labeled by “f” as in Figure 1. Take any spanning tree on the tree