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BALANCED PARTITIONS OF TWO SETS OF POINTS IN THE PLANE

Research Project

Project/Area Number 15540137
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionIBARAKI UNIVERSITY (2004)
Kogakuin University (2003)

Principal Investigator

KANO Mikio (2004)  IBARAKI UNIVERSITY, FACULTY OF ENGINEERING, PROFESSOR, 工学部, 教授 (20099823)

金子 篤司 (2003)  工学院大学, 工学部, 助教授 (30255608)

Co-Investigator(Kenkyū-buntansha) 加納 幹雄  茨城大学, 工学部, 教授 (20099823)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2003: ¥1,500,000 (Direct Cost: ¥1,500,000)
KeywordsBalanced Partition / Balanced subdivision / Red and blue points / Geometric graph / Discrete geometry / グラフ / 離散幾何学 / 平面 / 2種点集合 / 交差
Research Abstract

Recently, the Ham-sandwich Theorem was generalized as follows : If|R|=ag and|B|=bg, then there exists a subdivision X_1∪X_2∪・・・∪X_g of the plane into g disjoint convex polygons such that every X_i contains exactly a red points and b blue points. This theorem was proved by A. Kano and M.Kano for a=1,2 and they proposed it as a conjecture, and then this conjecture was independently proved by three papers.
We obtain the following three new results.
(1)Suppose that $R$ is a disjoint union of R_1 and R_2. Let|R_1|=g_1 and|R_2|=g_2. If|B|=(m-1)g_1+ mg_2, then we can subdivide the plane into g_1+g_2 disjoint convex polygons X_1∪・・・X_{g_1}∪Y_1∪・・・∪Y_{g_2} so that every X_i contains exactly one red point of R_1 and m-1 blue points, and every Y_j contains exactly one red point of R_2 and m blue points.
(2)Let a≧1, g≧0 and h≧0 be integers such that g+h≧1. If|R|=ag+(a+1)h and|B|=(a+1)g+ah, then there exists a subdivision X_1∪・・・∪X_g∪Y_1∪・・・∪ Y_h of the plane into g+h disjoint convex polygons such that every X_i contains exactly a red points and a+1 blue points and every Y_j contains exactly a+1 red points and a blue points.
(3)If|R|=a(g_1+g_2)+(a+1)g_3 and|B|=bg_1+(b+1)(g_2+g_3), then there exists a subdivision X_1∪・・・∪X_{g_1}∪Y_1∪・・・∪ Y_{g_2}∪Z_{1}∪・・・∪Z_{g_3} of the plane into g_1+g_2+g_3 disjoint convex polygons such that every X_i contains exactly a red points and b blue points, every Y_i, if any, contains exactly a red points and b+1 blue points, and every Z_i, if any, contains exactly a+1 red points and b+1 blue points
We also study some problems on discrete geometry and graph theory related to the above problems.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (17 results)

All 2005 2004 2003 Other

All Journal Article (16 results) Publications (1 results)

  • [Journal Article] A balanced interval of two sets of points on a line2005

    • Author(s)
      A.Kaneko, M.Kano
    • Journal Title

      Combinatorial Geometry and Graph Theory (LNCS) 3330

      Pages: 108-112

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Partitioning multipartite complete graphs by monochromatic trees2005

    • Author(s)
      A.Kaneko, M.Kano, K.Suzuki
    • Journal Title

      Journal Graph Theory 48

      Pages: 133-141

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A balanced interval of two sets of points on a line2005

    • Author(s)
      A.Kaneko, M.Kano
    • Journal Title

      Combinatorial Geometry and Graph Theory (Lecture Notes in Computer Science) Vol.3330

      Pages: 108-112

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A balanced interval of two sets of points on a line,2005

    • Author(s)
      A.Kaneko, M.Kano
    • Journal Title

      Combinatorial Geometry and Graph Theory(LNCS) 3330

      Pages: 108-112

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Partitioning multipartite complete graphs by monochromatic trees,2005

    • Author(s)
      A.Kaneko, M.Kano, K.Suzuki
    • Journal Title

      Journal Graph Theory 48

      Pages: 133-141

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Packing paths of lenghth at least two2004

    • Author(s)
      M.Kano, G.Katona, Z.Kiraly
    • Journal Title

      Discrete Mathematics 283

      Pages: 129-135

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Path Coverings of Two Sets of Points in the Plane2004

    • Author(s)
      A.Kaneko, M.Kano, K.Suzuki
    • Journal Title

      Towards a theory of geometric graphs, Contemporary Mathematics series of AMS 342

      Pages: 99-111

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Path Coverings of Two Sets of Points in the Plane2004

    • Author(s)
      A.Kaneko, M.Kano, K.Suzuki
    • Journal Title

      Towards a theory of geometric graphs (ed by J.Pach), Contemporary Mathematics series of AMS 342

      Pages: 99-111

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Packing paths of lenghth at least two,2004

    • Author(s)
      M.Kano, G.Katona, Z.Kiraly
    • Journal Title

      Discrete Mathematics 283

      Pages: 129-135

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Path Coverings of Two Sets of Points in the Plane,2004

    • Author(s)
      A.Kaneko, M.Kano, K.Suzuki
    • Journal Title

      Towards a theory of geometric graphs, Contemporary Mathematics series of AMS 342

      Pages: 99-111

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Discrete geometry on red and blue points in the plane --- A survey2003

    • Author(s)
      A.Kaneko, M.Kano
    • Journal Title

      Discrete and Computational Geometry, Algorithms Combin. 25

      Pages: 23-30

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Discrete geometry on red and blue points in the plane -A survey2003

    • Author(s)
      A.Kaneko, M.Kano
    • Journal Title

      Discrete and Computational Geometry, Algorithms Combin. 25

      Pages: 551-570

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Discrete geometry on red and blue points in the plane---A survey,2003

    • Author(s)
      A.Kaneko, M.Kano
    • Journal Title

      Discrete and Computational Geometry, Algorithms Combin. 25

      Pages: 23-30

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Semi-balanced partition of two sets of points and embedding of rooted forests

    • Author(s)
      A.Kaneko, M.Kano
    • Journal Title

      International Journal of Computational Geometry & Applications (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Semi-balanced partition of two sets of points and embedding of rooted forests

    • Author(s)
      A.Kaneko, M.Kano
    • Journal Title

      International Journal of Comutational Geometry & Appications (in print)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Semi-balanced partition of two sets of points and embedding of rooted forests,

    • Author(s)
      A.Kaneko, M.Kano
    • Journal Title

      International Journal of Computational Geometry & Applications 印刷中

    • Related Report
      2004 Annual Research Report
  • [Publications] A.Kaneko, M.Kano, K.Suzuki: "Path coverings of two sets of points in the plane"Towards a Theory of Geometric Graphs (Contemporary Mathematics (American Mathematical Society)). Vol 342. 56-63 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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