THE RAMSEY NUMBER FOR A LINEAR FOREST VERSUS TWO IDENTICAL COPIES OF COMPLETE GRAPHS

Abstract

Let H be a graph with the chromatic number h and the chromatic surplus s. A connected graph G of order n is called H-good if R(G, H) = (n - 1)(h - 1) + s. In this paper, we show that Pn is 2Km-good for n ≥ 3. Furthermore, we obtain the Ramsey number R(L, 2Km), where L is a linear forest. Moreover, we also give the Ramsey number R(L, Hm) which is an extension for R(kPn, Hm) proposed by Ali et al. [1], where Hm is a cocktail party graph on 2m vertices.