Hiroshi Maehara

Is There a Circle that Passes Through a Given Number of Lattice Points

It is proved that for any integern?0, there is a circle in the plane that passes through exactlynlattice points.

On the Number of Acute Triangles in a Straight-Line Embedding of a Maximal Planar Graph

In this paper we show that any maximal planar graph withmtriangles except the unbounded face can be transformed into a straight-line embedding in which at least ?m/3? triangles are acute triangles. Moreover, we show that any maximal outer-planar graph can be transformed into a straight-line embedding in which all faces are acute triangles except the unbounded face.

ON RANDOM SIMPLICES IN PRODUCT DISTRIBUTIONS

It is proved that the r-dimensional volume V. of the r-simplex spanned by r + 1 sample points taken from the n-fold product distribution of a distribution F is asymptotically normal as n -* oo. RANDOM SIMPLEX; PRODUCT DISTRIBUTION; ASYMPTOTIC NORMALITY OF VOLUME

On Knotted Necklaces of Pearls

A knotted necklace of pearls is a necklace whose string forms a nontrivial knot. We assume that all pearls are balls of the same size, and any two consecutive pearls in a knotted necklace are tangent to each other. How many pearls are necessary to make a knotted necklace, say a trefoil knot? We show that 15 pearls are sufficient, and that to make a knotted necklace which can be put in the shallowest-possible show-case (a box with a glass-lid), 16 pearls are necessary and sufficient.

Subdividing a Graph Toward a Unit-distance Graph in the Plane

The subdivision number of a graph G is defined to be the minimum number of extra vertices inserted into the edges of G to make it isomorphic to a unit-distance graph in the plane. Lett (n) denote the maximum number of edges of a C4-free graph on n vertices. It is proved that the subdivision number of Knlies betweenn (n? 1)/2 ?t(n) and (n? 2)(n? 3)/2 + 2, and that of K(m, n) equals (m? 1)(n?m) forn?m(m? 1).

Cutting a Set of Disks by a Line with Leaving Many Intact Disks in Both Sides

Cut by a line the union of given disjoint disks in the plane so that both sides of the line contain many intact disks. At least how many intact disks can we leave in either side? It is proved that there is a family of infinitely many disjoint disks in the plane for which every line has a side that contains at most one intact disk. On the other hand, for any family of n disjoint disks, there is a circle C such that both the interior and the exterior of C contain n/4?o(n) intact disks.

Can a convex polyhedron have a developable face-cycle?

Abstract A face-cycle of a polyhedron in R 3 is a cyclic sequence of at least three, distinct faces of the polyhedron such that each consecutive pair of faces have a common edge. We prove that for any face-cycle cut out from a (cardboard) convex polyhedron, it is impossible to unfold the face-cycle to lie flat on the plane.

Configuration Spaces of Pentagonal Frameworks

Let F be a pentagonal framework in the plane. When we deform F continuously in the plane, the shape of F changes. The configuration space of F is the space of its all possible `shapes?. We characterize and classify the configuration spaces for those pentagonal frameworks that cannot be folded into a line.

Embedding a set of rational points in lower dimensions

Abstract Let X n be a set of rational points lying on an n -dimensional flat in a Euclidean space. We prove that for n ⩾ 2, X n is congruent to a set of rational points in R 2n+1 , and that for n ⩾ 3, X n is similar to a set of rational points in R 2n-1 .

Piercing Balls Sitting on a Table by a Vertical Line

Let Fnbe a family of disjoint n balls all sitting on a fixed horizontal table T. Let ? denote a vertical line that meets T. We prove that if ? meets 2 k+ 1 balls in Fn, then the radius of the smallest ball among the 2k+ 1 balls is at most (2 ?3)ktimes the radius of the biggest ball among the 2 k+ 1 balls. Using this result we prove that for anyFn the average number of balls an ? meets is at most logn+o(1). A similar result for a two-dimensional version is also given together with a lower bound of the least upper bound.