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Algorithmica

, Volume 57, Issue 4, pp 869–872 | Cite as

The Steiner Ratio Conjecture of Gilbert-Pollak May Still Be Open

  • N. Innami
  • B. H. Kim
  • Y. Mashiko
  • K. Shiohama
Article

Abstract

We offer evidence in the disproof of the continuity of the length of minimum inner spanning trees with respect to a parameter vector having a zero component. The continuity property is the key step of the proof of the conjecture in Du and Hwang (Proc. Nat. Acad. Sci. U.S.A. 87:9464–9466, 1990; Algorithmica 7(1):121–135, 1992). Therefore the Steiner ratio conjecture proposed by Gilbert-Pollak (SIAM J. Appl. Math. 16(1):1–29, 1968) has not been proved yet. The Steiner ratio of a round sphere has been discussed in Rubinstein and Weng (J. Comb. Optim. 1:67–78, 1997) by assuming the validity of the conjecture on a Euclidean plane in Du and Hwang (Proc. Nat. Acad. Sci. U.S.A. 87:9464–9466, 1990; Algorithmica 7(1):121–135, 1992). Hence the results in Rubinstein and Weng (J. Comb. Optim. 1:67–78, 1997) have not been proved yet.

Keywords

Steiner ratio Gilbert-Pollak conjecture Steiner trees Spanning trees 

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References

  1. 1.
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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • N. Innami
    • 1
  • B. H. Kim
    • 2
  • Y. Mashiko
    • 3
  • K. Shiohama
    • 4
  1. 1.Department of Mathematics, Faculty of ScienceNiigata UniversityNiigataJapan
  2. 2.Department of MathematicsKyung Hee UniversitySuwonKorea
  3. 3.Department of Mathematics, Faculty of Science and EngineeringSaga UniversitySagaJapan
  4. 4.Department of Applied Mathematics, Faculty of SciencesFukuoka UniversityFukuokaJapan

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