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Any set Pn of n points can be convex-quadrangulated with at most 4n 5 + 2 Steiner points located in the interior of Conv(Pn ). 1 Lower Bounds In [6] it was proved that there are families of point sets (not in general position) with n points for which n−3 − 1 Steiner points are needed to convex2 quadrangulate them. For points in general position, an example is also presented On Convex Quadrangulations of Point Sets on the Plane 45 in which n4 points are necessary. The n4 lower bound can be improved as follows: Consider a convex polygon Q with an even number of vertices, and for every other edge of Q, place a point in the interior of Q at distance from the middle point of the edge.

Two cases arise: p1 lies in the interior of Conv(Pn ), or p1 is a vertex of Conv(Pn ). We begin quadrangulating as shown in Figure 2(b). Observe that in the first case, we insert a Steiner point slightly below the line segment joining p to p1 , and in the second we place a Steiner point slightly above On Convex Quadrangulations of Point Sets on the Plane 41 the line joining p to p1 and on the boundary of Conv (Pn ). We now proceed inductively, assuming that if p2(i−1)+1 = p2i−1 is a vertex, there is a Steiner point slightly below the line joining p to p2i−1 , or if p2i−1 is an interior point to Conv (Pn ) then there is a Steiner point on the boundary of Conv (Pn ) slightly above the line segment joining p to p2i−1 , i ≥ 0.

Li [10]) Let G be a 2-connected weighted graph and m a nonnegative number. Then G contains either a Hamilton cycle or a cycle of weight at least m if it satisfies the following conditions: (1) max{idw (u), idw (v)|d(u, v) = 2} ≥ m/2; (2) w(ux) = w(vx) for every vertex x ∈ N (u) ∩ N (v) with d(u, v) = 2; (3) In every triangle T of G, either all the edges of T have different weights or all the edges of T have the same weight. 24 B. Chen, S. E. Cheng Fujisawa [9] gave the so-called claw conditions for the existence of heavy cycles in weighted graphs, generalizing Theorem C.

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