• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

A challenge of solving the normality conjecture for cut polytopes affirmatively which yields a theoretical proof of the four color theorem

Research Project

Project/Area Number 25610032
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionOsaka University

Principal Investigator

HIBI Takayuki  大阪大学, 情報科学研究科, 教授 (80181113)

Project Period (FY) 2013-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2015: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords切断多面体 / 切断イデアル / 正規多面体 / 正則単模三角形分割 / トーリックイデアル / グレブナー基底 / イニシャルイデアル / 実験計画 / 2水準一部実験計画 / Hilbert basis / 2水準一部実施計画
Outline of Final Research Achievements

The purpose of the present research was, following the established techniques due to the principal investigator together with his colleagues, to face a challenge of solving the normality conjecture for cut polytopes affirmatively. The reason why the normality conjecture for cut polytopes fascinates the combinatorialists is the fact, due to David E. Speyer, that the affirmative answer of the normality conjecture for cut polytopes guarantees the four color theorem. We have faced a challenge of solving the normality conjecture for cut polytopes affirmatively from the viewpoint of combinatorics on dilated cut polytopes as well as that of initial ideals of Gr\"obner bases of cut ideals. Furthermore, with taking into account the negative answer of the normality conjecture for cut polytopes, we have developed the various studies on cut polytopes and cut ideals.

Report

(4 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (19 results)

All 2015 2014 2013 Other

All Journal Article (12 results) (of which Peer Reviewed: 12 results,  Open Access: 2 results) Presentation (3 results) (of which Int'l Joint Research: 1 results,  Invited: 3 results) Book (1 results) Remarks (2 results) Funded Workshop (1 results)

  • [Journal Article] Centrally symmetric configurations of order polytopes2015

    • Author(s)
      T. Hibi, K. Matsuda, H. Ohsugi, K. Shibata
    • Journal Title

      Journal of Algebra

      Volume: 443 Pages: 469-478

    • DOI

      10.1016/j.jalgebra.2015.06.010

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Many toric ideals generated by quadratic binomials possess no quadratic Groebner bases2014

    • Author(s)
      T. Hibi, K. Nishiyama, H. Ohsugi and A. Shikama
    • Journal Title

      J. Algebra

      Volume: 408 Pages: 138-146

    • DOI

      10.1016/j.jalgebra.2013.09.039

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Toric ideals of finite graphs and adjacent 2-minors2014

    • Author(s)
      H. Ohsugi and T. Hibi
    • Journal Title

      Math. Scand.

      Volume: 114

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Centrally symmetric configurations of integer matrices2014

    • Author(s)
      H. Ohsugi and T. Hibi
    • Journal Title

      Nagoya Mathematical Journal

      Volume: 216 Pages: 153-170

    • DOI

      10.1215/00277630-2857555

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Integer decomposition property of dilated polytopes2014

    • Author(s)
      D. A. Cox, C. Haase, T. Hibi and A. Higashitani
    • Journal Title

      Electron. J. Combin.

      Volume: 21

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Edge rings satisfying Serre's condition (R_1)2014

    • Author(s)
      T. Hibi and L. Katthaen
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: 142 Issue: 7 Pages: 2537-2541

    • DOI

      10.1090/s0002-9939-2014-11973-3

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Depth of initial ideals of normal edge rings2014

    • Author(s)
      T. Hibi, A. Higashitani, K. Kimura, Kyouko and A. O'keefe
    • Journal Title

      Communications in Algebra

      Volume: 42 Issue: 7 Pages: 2908-2922

    • DOI

      10.1080/00927872.2012.760565

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Monomial ideals whose depth function has any given number of strict local maxima2014

    • Author(s)
      S. Bandari, J. Herzog and T. Hibi
    • Journal Title

      Ark. Mat.

      Volume: 52 Issue: 1 Pages: 11-19

    • DOI

      10.1007/s11512-013-0184-1

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] The binomial edge ideal of a pair of graphs2014

    • Author(s)
      V. Ene, J. Herzog, T. Hibi and A. A. Qureshi
    • Journal Title

      Nagoya Math. J.

      Volume: 213 Pages: 105-125

    • DOI

      10.1215/00277630-2389872

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Toric ideals and their circuits2013

    • Author(s)
      H. Ohsugi and T. Hibi
    • Journal Title

      J. Commutative Algebra

      Volume: 5 Issue: 2 Pages: 309-322

    • DOI

      10.1216/jca-2013-5-2-309

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] The join-meet ideal of a finite lattice2013

    • Author(s)
      V. Ene and T. Hibi
    • Journal Title

      J. Commutative Algebra

      Volume: 5 Issue: 2

    • DOI

      10.1216/jca-2013-5-2-209

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Markov chain Monte Carlo methods for the regular two-level fractional factorial designs and cut ideals2013

    • Author(s)
      S. Aoki, T. Hibi and H. Ohsugi
    • Journal Title

      J. Statist. Plann. Inference

      Volume: 143 Issue: 10 Pages: 1791-1806

    • DOI

      10.1016/j.jspi.2013.06.009

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Presentation] Gorenstein Fano polytopes and quadratic Gr\"obner bases2015

    • Author(s)
      T. Hibi
    • Organizer
      Combinatorial and Experimental Methods in Commutative Algebra and Related Fields
    • Place of Presentation
      Universitaet Osnabrueck(ドイツ)
    • Year and Date
      2015-10-08
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Stanley's Influence on Monomial Ideals2014

    • Author(s)
      T. Hibi
    • Organizer
      A conference in honor of Richard P. Stanley’s 70th birthday
    • Place of Presentation
      Massachusetts Institute of Technology
    • Year and Date
      2014-06-23 – 2014-06-27
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Cameron--Walker graphs2013

    • Author(s)
      T. Hibi
    • Organizer
      International conference on Commutative Algebra and its Interaction to Algebraic Geometry and Combinatorics
    • Place of Presentation
      Hanoi, Vietnam
    • Related Report
      2013 Research-status Report
    • Invited
  • [Book] Groebner Bases: Statistics and Software Systems2013

    • Author(s)
      T. Hibi, Ed.
    • Total Pages
      474
    • Publisher
      Springer
    • Related Report
      2013 Research-status Report
  • [Remarks] 日比孝之|研究科紹介|大阪大学大学院情報科学研究科

    • URL

      http://www.ist.osaka-u.ac.jp/japanese/introduction/professors/pure-and-applied-mathematics/combinatorics/takayuki-hibi.html

    • Related Report
      2015 Annual Research Report
  • [Remarks] 大阪大学大学院情報科学研究科情報基礎数学専攻組合せ数学講座教員紹介

    • URL

      http://www.ist.osaka-u.ac.jp/japanese/introduction/professors/pure-and-applied-mathematics/combinatorics/takayuki-hibi.html

    • Related Report
      2014 Research-status Report
  • [Funded Workshop] グレブナー基底の50年2015

    • Place of Presentation
      ホテル日航大阪(大阪)
    • Year and Date
      2015-07-01
    • Related Report
      2015 Annual Research Report

URL: 

Published: 2014-07-25   Modified: 2019-07-29  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi