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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1191, author = {Ralph Faudree and Annette Schelten and Ingo Schiermeyer}, title = {The Ramsey number r(C7,C7,C7)}, journal = {Discussiones Mathematicae Graph Theory}, volume = {23}, year = {2003}, pages = {141-158}, zbl = {1049.05055}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1191} }
Ralph Faudree; Annette Schelten; Ingo Schiermeyer. The Ramsey number r(C₇,C₇,C₇). Discussiones Mathematicae Graph Theory, Tome 23 (2003) pp. 141-158. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1191/
[000] [1] A. Bialostocki and J. Schönheim, On Some Turan and Ramsey Numbers for C₄, Graph Theory and Combinatorics, Academic Press, London, (1984) 29-33. | Zbl 0554.05036
[001] [2] J.A. Bondy and P. Erdős, Ramsey Numbers for Cycles in Graphs, J. Combin. Theory (B) 14 (1973) 46-54. | Zbl 0248.05127
[002] [3] S. Brandt, A Sufficient Condition for all Short Cycles, Discrete Applied Math. 79 (1997) 63-66. | Zbl 0882.05081
[003] [4] S. Brandt and H.J. Veldman, Degree sums for edges and cycle lengths in graphs, J. Graph Theory 25 (1997) 253-256. | Zbl 0876.05056
[004] [5] V. Chvátal, On Hamiltonian's Ideals, J. Combin. Theory (B) 12 (1972) 163-168.
[005] [6] C. Clapham, The Ramsey Number r(C₄,C₄,C₄), Periodica Mathematica Hungarica 18 (1987) 317-318.
[006] [7] P. Erdős, On the Combinatorial Problems which I would most Like to See Solved, Combinatorica 1 (1981) 25-42. | Zbl 0486.05001
[007] [8] R.J. Faudree and R.H. Schelp, All Ramsey Numbers for Cycles in Graphs, Discrete Math. 8 (1974) 313-329. | Zbl 0294.05122
[008] [9] R.E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian J. Math. 7 (1995) 1-7. | Zbl 0064.17901
[009] [10] T. Łuczak, R(Cₙ,Cₙ,Cₙ) ≤ (4+o(1))n, J. Combin. Theory (B) 75 (1999) 174-187.
[010] [11] S.P. Radziszowski, Small Ramsey Numbers, Electronic J. Combin. 1 (1994) update 2001.
[011] [12] A. Schelten, Bestimmung von Ramsey-Zahlen zweier und dreier Graphen (Dissertation, TU Bergakademie Freiberg, 2000).
[012] [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. | Zbl 0756.05078
[013] [14] P. Rowlinson and Yang Yuangsheng, On Graphs without 6-Cycles and Related Ramsey Numbers, Utilitas Mathematica 44 (1993) 192-196. | Zbl 0789.05070