Ramsey Numbers for the Pair Sparse Graph-path or Cycle

Let G be a connected graph on n vertices with no more than n(\ + 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 e is sufficiently small then for k odd r(G, Ck) In 1. Also, for k > 2, r(G, Pk) = max{/i -t[k/2] l,n + k 2 a' 8), where a' is the independence number of an appropriate subgraph of G and 5 is 0 or 1 depending upon n, k and a'. Introduction. Let G and 77 be simple graphs. The Ramsey number r(G, H) is the smallest integer « such that for each graph F on « vertices, either G is a subgraph of F or 77 is a subgraph of F, the complement of F. Calculation of r(G, H) for particular pairs of graphs G and 77 has received considerable attention, and a survey of such results can be found in [2]. Chvátal [5] proved that if Tn is a tree on « vertices and Km is a complete graph on m vertices, then r(Tn, Km) = (n — l)(m — 1) + 1. In [4] it was shown that if Tn is replaced by a sparse connected graph Gn on « vertices the Ramsey number remains the same (i.e. r(G„, Km) = (n — l)(m — 1) 41). For m = 3 Chvátal's theorem implies r(Tn, K3) = 2n — I. In this paper we will show that if Tn is replaced by any sparse connected graph G on « vertices and K3 is replaced by an odd cycle Ck, then for appropriate « the Ramsey number is unchanged. In particular we will prove the following. Theorem. If G is a connected graph on n vertices and no more than «(1 + e) edges, then