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Graphs and Combinatorics

, Volume 27, Issue 1, pp 121–128 | Cite as

Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs

  • Tao JiangEmail author
  • Michael Salerno
Original Paper

Abstract

The Ramsey number r(H, K n ) is the smallest positive integer N such that every graph of order N contains either a copy of H or an independent set of size n. The Turán number ex(m, H) is the maximum number of edges in a graph of order m not containing a copy of H. We prove the following two results: (1) Let H be a graph obtained from a tree F of order t by adding a new vertex w and joining w to each vertex of F by a path of length k such that any two of these paths share only w. Then \({r(H,K_n)\leq c_{k,t}\, {n^{1+1/k}\over \ln^{1/k} n}}\) , where c k,t is a constant depending only on k and t. This generalizes some results in Li and Rousseau (J Graph Theory 23:413–420, 1996), Li and Zang (J Combin Optim 7:353–359, 2003), and Sudakov (Electron J Combin 9, N1, 4 pp, 2002). (2) Let H be a bipartite graph with ex(m, H) = O(m γ ), where 1 < γ < 2. Then \({r(H,K_n)\leq c_H ({n\over \ln n})^{1/(2-\gamma)}}\), where c H is a constant depending only on H. This generalizes a result in Caro et al. (Discrete Math 220:51–56, 2000).

Keywords

Ramsey number Independence number 

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Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Department of MathematicsMiami UniversityOxfordUSA

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