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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
  • 54 Downloads

Abstract

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.

Keywords

Ramsey number Path Star Tree Generalized wheel 

Notes

Acknowledgements

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.

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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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