The cycle-complete graph Ramsey number r (Cm,Kn ) is the smallest integer N such that every graph G of order N contains a cycle Cm on m vertices or has independence number (G) n. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that r (Cm,Kn )1⁄4 (m 1) (n 1)þ 1 for all m n 3 (except r (C3,K3 )1⁄4 6). This conjecture holds for 3 n 5: In this paper we will present a proof for n 1⁄4 ...

The Ramsey number r(H, G) is defined as the minimum N such that for any coloring of the edges of the N-vertex complete graph KN in red and blue, it must contain either a ted H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K,, G)< 2qf 1 where G has q edges. In other words, any graph on 2q+ 1 vertices with independence number at most 2 contains every (isolat...

A multiple-interval representation of a graph G is a mapping f which assigns to each vertex of G a union of intervals on the real line so that two distinct vertices u and v are adjacent if and only if f(u) ∩ f(v) = ∅. We study the total interval number of G, de0ned as I(G) = min {∑ v∈V #f(v): f is a multiple-interval representation of G } ; where #f(v) is the minimum number of intervals whose u...