Ramsey numbers for the pair sparse graph-path or cycle

Abstract

Abstract. 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,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) 4-1). 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

Ramsey numbers of trees and unicyclic graphs versus fans

“…have not yet been embedded to, if either t 1 or t 2 has not yet been embedded. Furthermore, map C 1 ∪ C 2 ∪ · · · ∪ C t − {y 1 , y 2 , .…”

“…10 [2]. Salman and Broersma determined the Ramsey number of paths versus fans, finding R(P n , F m ) for various ranges of n and m [10].…”

Goodness of trees for generalized books

“…, .,yh) be disjoint sets of vertices of a complete graph which is (R, B) a-colored. (ii) [7] There is either a blue matching of X into Y or for some c ( Theorem C. [2] Let T be a tree with n vertices. If T has no suspended path with more than a vertices, then G has at least rnl2al vertices of degree one.…”

Generalized Ramsey numbers for graphs

“…Note. It was proved by Burr et al [21] that R(T m , C n ) = 2m − 1 for odd n and m ≥ 756n 10 . Thus, R(T m ,W n ) = 3m − 2 for odd n and m ≥ 756n 10 .…”

“…Another natural question is for what graphs G, T n is G-good. Theorem 5.2 (Burr et al [21]). R(T n , C m ) = 2n − 1 for odd m ≥ 3 and n ≥ 756m 10 .…”