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RAMSEY NUMBERS FOR TREES

  • ZHI-HONG SUN (a1)
Abstract
Abstract

For n≥5, let Tn denote the unique tree on n vertices with Δ(Tn)=n−2, and let T*n=(V,E) be the tree on n vertices with V ={v0,v1,…,vn−1} and E={v0v1,…,v0vn−3,vn−3vn−2,vn−2vn−1}. In this paper, we evaluate the Ramsey numbers r(Gm,Tn) and r(Gm,T*n) , where Gm is a connected graph of order m. As examples, for n≥8 we have r(Tn,T*n)=r(T*n,T*n)=2n−5 , for n>m≥7 we have r(K1,m−1,T*n)=m+n−3 or m+n−4 according to whether m−1∣n−3 or m−1∤n−3 , and for m≥7 and n≥(m−3)2 +2 we have r(T*m,T*n)=m+n−3 or m+n−4 according to whether m−1∣n−3 or m−1∤n−3 .

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[1] S. A. Burr , ‘Generalized Ramsey theory for graphs—a survey’, in: Graphs and Combinatorics, Lecture Notes in Mathematics, 406 (eds. R.A. Bari and F. Harary ) (Springer, Berlin–New York, 1974), pp. 5275.

[3] P. Erdős and T. Gallai , ‘On maximal paths and circuits in graphs’, Acta Math. Acad. Sci. Hungar. 10 (1959), 337356.

[4] G. H. Fan and L. L. Sun , ‘The Erdős–Sós conjecture for spiders’, Discrete Math. 307 (2007), 30553062.

[5] R. J. Faudree and R. H. Schelp , ‘Path Ramsey numbers in multicolorings’, J. Combin. Theory Ser. B 19 (1975), 150160.

[9] A. F. Sidorenko , ‘Asymptotic solution for a new class of forbidden r-graphs’, Combinatorica 9 (1989), 207215.

[11] M. Woźniak , ‘On the Erdős–Sós conjecture’, J. Graph Theory 21 (1996), 229234.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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