Designs, Codes and Cryptography

, Volume 62, Issue 3, pp 323–330 | Cite as

Steiner triple systems satisfying the 4-vertex condition

  • Petteri Kaski
  • Mahdad Khatirinejad
  • Patric R. J. Östergård
Article

Abstract

Higman asked which block graphs of Steiner triple systems of order v satisfy the 4-vertex condition and left the cases v = 9, 13, 25 unsettled.We give a complete answer to this question by showing that the affine plane of order 3 and the binary projective spaces are the only such systems. The major part of the proof is to show that no block graph of a Steiner triple system of order 25 satisfies the 4-vertex condition.

Keywords

4-Vertex condition Steiner triple system Strongly regular graph 

Mathematics Subject Classification (2000)

05B07 05E30 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Petteri Kaski
    • 1
  • Mahdad Khatirinejad
    • 2
    • 3
  • Patric R. J. Östergård
    • 2
  1. 1.Department of Information and Computer Science, Helsinki Institute of Information Technology HIITAalto UniversityAaltoFinland
  2. 2.Department of Communications and NetworkingAalto UniversityAaltoFinland
  3. 3.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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