Download PDF - The Ramsey number r(C₇,C₇,C₇)All rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

The Ramsey number r(C₇,C₇,C₇)

Authors 1, 2, 2

Affiliations

  1. Department of Mathematical Science, University of Memphis, Memphis, TN 38152, USA
  2. Fakultät für Mathematik und Informatik, Technische Universität Bergakademie Freiberg, 09596 Freiberg, Germany

Abstract

Bondy and Erdős [2] have conjectured that the Ramsey number for three cycles Cₖ of odd length has value r(Cₖ,Cₖ,Cₖ) = 4k-3. We give a proof that r(C₇,C₇,C₇) = 25 without using any computer support.

Keywords

ENG
Ramsey numbers, extremal graphs

Bibliography

  1. A. Bialostocki and J. Schönheim, On Some Turan and Ramsey Numbers for C₄, Graph Theory and Combinatorics, Academic Press, London, (1984) 29-33.
  2. J.A. Bondy and P. Erdős, Ramsey Numbers for Cycles in Graphs, J. Combin. Theory (B) 14 (1973) 46-54.
  3. S. Brandt, A Sufficient Condition for all Short Cycles, Discrete Applied Math. 79 (1997) 63-66.
  4. S. Brandt and H.J. Veldman, Degree sums for edges and cycle lengths in graphs, J. Graph Theory 25 (1997) 253-256.
  5. V. Chvátal, On Hamiltonian's Ideals, J. Combin. Theory (B) 12 (1972) 163-168.
  6. C. Clapham, The Ramsey Number r(C₄,C₄,C₄), Periodica Mathematica Hungarica 18 (1987) 317-318.
  7. P. Erdős, On the Combinatorial Problems which I would most Like to See Solved, Combinatorica 1 (1981) 25-42.
  8. R.J. Faudree and R.H. Schelp, All Ramsey Numbers for Cycles in Graphs, Discrete Math. 8 (1974) 313-329.
  9. R.E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian J. Math. 7 (1995) 1-7.
  10. T. Łuczak, R(Cₙ,Cₙ,Cₙ) ≤ (4+o(1))n, J. Combin. Theory (B) 75 (1999) 174-187.
  11. S.P. Radziszowski, Small Ramsey Numbers, Electronic J. Combin. 1 (1994) update 2001.
  12. A. Schelten, Bestimmung von Ramsey-Zahlen zweier und dreier Graphen (Dissertation, TU Bergakademie Freiberg, 2000).
  13. P. Rowlinson amd Yang Yuangsheng, On the Third Ramsey Numbers of Graphs with Five Edges, J. Combin. Math. and Combin. Comp. 11 (1992) 213-222.
  14. P. Rowlinson and Yang Yuangsheng, On Graphs without 6-Cycles and Related Ramsey Numbers, Utilitas Mathematica 44 (1993) 192-196.
Discussiones Mathematicae Graph Theory
2003/23/1
Export to PBN, Opens in new tab
Pages:
141-158
Main language of publication
English
Received
2001-07-30
Accepted
2002-01-18
Published
2003

DOI: 10.7151/dmgt.1191

Exact and natural sciences