Graphs and Combinatorics

, Volume 35, Issue 1, pp 189–193 | Cite as

The Ramsey Numbers of Trees Versus Generalized Wheels

  • Longqin Wang
  • Yaojun Chen
Original Paper


For two given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1,G_2)\) is the smallest integer n such that for any graph G of order n, either G contains \(G_1\) or its complement \({\overline{G}}\) contains \(G_2\). Let \(P_n, S_n\) and \(T_n\) denote a path, a star and a tree of order n, respectively. A generalized wheel, denoted by \(W_{s,m}\), is the join of a complete graph \(K_s\) and a cycle \(C_m\). In this paper, we show that \(R(T_n,W_{s,4})=(n-1)(s+1)+1\) for \(n\ge 3,s\ge 2\) and \(R(T_n,W_{s,5})=(n-1)(s+2)+1\) for \(n\ge 3,s\ge 1\). These generalize some known results on Ramsey numbers for a tree versus a wheel.


Ramsey number Path Star Tree Generalized wheel 



We are grateful to the anonymous referees for their many careful comments on our earlier version of this paper. This research was supported by NSFC under Grant numbers 11671198 and 11871270.


  1. 1.
    Baskoro, E.T., Nababan, S.M., Miller, M.: On Ramsey numbers for trees versus wheels of five or six vertices. Graphs Comb. 18, 717–721 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bondy, J.A.: Pancyclic graphs. J. Comb. Theory Ser. B 11, 80–84 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burr, S.A.: Ramsey numbers involving graphs with long suspended paths. J. Lond. Math. Soc. 24, 405–413 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, Y., Zhang, Y., Zhang, K.: The Ramsey numbers of paths versus wheels. Discrete Math. 290, 85–87 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Y., Zhang, Y., Zhang, K.: The Ramsey numbers of stars versus wheels. Eur. J. Comb. 25, 1067–1075 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hasmawati, H., Baskoro, E.T., Assiyatun, H.: Star-wheel Ramsey numbers. J. Comb. Math. Comb. Comput. 55, 123–128 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Lin, Q., Li, Y., Dong, L.: Ramsey goodness and generalized stars. Eur. J. Comb. 31, 1228–1234 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Radziszowski, S.P.: Small Ramsey numbers. Electron. J. Comb. DS1, 15 (2017)MathSciNetGoogle Scholar
  9. 9.
    Radziszowski, S.P., Xia, J.: Paths, cycles and wheels without antitriangles. Australas. J. Comb. 9, 221–232 (1994)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Surahmat, E.T.: Baskoro, On the Ramsey number of path or star versus \(W_4\) or \(W_5\). In: Proceedings of the 12th Australasian Workshop on Combinatorial Algorithms, Bandung, 14–17 July 2001, pp. 174–179 (2001)Google Scholar
  11. 11.
    Zhang, Y.: On Ramsey numbers of short paths versus large wheels. ARS Comb. 89, 11–20 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouPeople’s Republic of China

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