### Ramsey numbers for the pair sparse graph-path or cycle

S. A. Burr, P. Erdős, R. J. Faudree, C. C. Rousseau, R. H. Schelp
1982 Transactions of the American Mathematical Society
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
more » ... ber r(G, H) is the smallest integer n such that for each graph F on n vertices, either G is a subgraph of F or H is a subgraph of F, the complement of F . Calculation of r(G, H) for particular pairs of graphs G and H has received considerable attention, and a survey of such results can be found in  . Chvátal  proved that if T" is a tree on n vertices and K", is a complete graph on m vertices, then r(T", Km ) _ (n -1)(m -1) + 1 . In  it was shown that if T" is replaced by a sparse connected graph G" on n vertices the Ramsey number remains the same (i .e. r(G", K_ (n -1)(m -1) + 1) . For m = 3 Chvátal's theorem implies r(T", K3 ) = 2n -1 . In this paper we will show that if T" is replaced by any sparse connected graph G on n vertices and K3 is replaced by an odd cycle Ck, then for appropriate n 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 n(1 + e) edges, then r(G,Ck )=2n-1 for n sufficiently large, e sufficiently small (both depending upon k) and k odd . This theorem falls into a larger category of results considered by Burr  . A