| Co-Investigator(Kenkyū-buntansha) |
ITOH Yoshiaki The Institute of Statistical Mathematics, Professor, 教授 (60000212)
ENOMOTO Hikoe Keio Univ., Dept.of Mathematics, Professor, 理工学部, 教授 (00011669)
KANO Mikio Ibaraki Univ., Dept.of Computer and Information Science, Professor, 工学部, 教授 (20099823)
OZEKI Michio Yamagata Univ., Faculty of Science, Professor, 理学部, 教授 (90087073)
TOKUSHIGE Norihide Univ.of the Ryukyus, College of Education, Associate Professor, 教育学部, 助教授 (00217481)
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| Research Abstract |
1.On the rigidity of frameworks : We presented a rigid unit-bar-framework in the 3-dimensional space that has no triangle, and a minimum rigid framework in the plane that cannot be constructed from the data of edge-lengths and graph structure. We proved that if a complete bipartite framework K (m, n) (m >= 3, n >= 5) in the plane admits a continuous deformation, then one of the partite-sets lies on a line L and the other partite-set lies on the line perpendicular to L.
2.On embeddings of structures : We proved that for any planar graph G = (V,E), there is an emebedding f : V * R^2 such that x, y * V are adijacent if and only if the distance between f (x) and f (y) is an integer. We also proved that every n dimensional inner product space over the rational field can be isometrically embedded into 2n + 1 dimensional Euclidean space.
3.On srrangements of spheres : A graph G is said to be representable by balls on the table, if we can place solid balls on a table, one ball for each vertex, so that two balls are tangent only when the corresponding vertices are adjacent. We proved that the family F of graphs representable by balls on a table is different from the family of planar graphs, and that F does not contain any member of the so-called Petersen family, and that the maximum value of the chromatic number of a graph in F is either 5 or 6.
4.On random graphs, probability : We determined the probability distribution of the order of the maximum regular tournament in a dominance relation generated by a randam n points on a circle. We extended the classical ruin problem to 3 persons' game, and calculated the probability that a fixed gambler A ruined first, and the probability that A is the sole survivor.
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