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Ramsey numbers for the pair sparse graph-path or cycle

1982
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Transactions of the American Mathematical Society
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ABSTRACr . Let G be a connected graph on n vertices with no more than n(1 + e) edges, and Pk or Ck a path or cycle with k vertices . In this paper we will show that if n is sufficiently large and a is sufficiently small then for k odd r(G, Ck ) = 2n -1 . Also, for k > 2, r(G,Pk )=max{n+[k/2]-1,n+k-2-á -ő}, where á is the independence number of an appropriate subgraph of G and ő is 0 or 1 depending upon n, k and a' . Introduction . Let G and H be simple graphs . The Ramsey number r(G, H) is the

doi:10.1090/s0002-9947-1982-0637704-5
fatcat:pptfyfhpgvet5p3lzwl6s4b4pi