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.

A note on the Ramsey number for cycle with respect to multiple copies of wheels

“…In particular, when G is a disjoint union of cycles then by using the results of Sudarsana et al [11] and Theorem 1.1 we obtain the Corollary 1.1 below for finding the exact value of Ramsey number R(G, tW 4 ).…”

“…Let G be a graph containing all H-good components, the general formula for finding the exact value of Ramsey number R(G, H) have been found by Bielak [2] and Sudarsana et al [11]. In some particular graphs, showed by Stahl [10] and Baskoro et al [1].…”

“…The disjoint union of graphs, t i=1 G i , is a graph with the vertex set t i=1 V i and the edge set t i=1 E i ; and if G i G for each i then t i=1 G i = tG. For the definition of Ramsey number, we cite the results proposed by Sudarsana [13], [11], [12]; that is, for graphs G and H, the Ramsey number R(G, H) is the smallest natural number n such that in every red and blue colorings of the edges of the complete graph K n , there is a graph G in K n which is all edges are red or a graph H in K n which is all edges are blue. In the other terminology, a graph F is called (G, H)-free, if F contains no G and F contains no H. Furthermore, we have an equivalent definition of Ramsey number R(G, H), that is, the smallest positive integer n such that there is no (G, H)-free graph on n vertices exists.…”

The Ramsey Number for a Forest versus Disjoint Union of Complete Graphs

“…Results without the assumption of sufficiently large number of vertices are also interesting. In [15], Sudarsana, Adiwijaya and Musdalifah showed that P n is 2K mgood for n ≥ 3 and m ≥ 2, and conjectured that any tree T n with n vertices is 2K m -good. Recently, Pokrovskiy and Sudakov [11] proved that for a fixed graph H, the path on n vertices with n ≥ 4v(H) is H-good.…”

“…By applying this result, Stahl [13] determined the Ramsey number of a forest versus Km. Concerning whether a tree is H-good for H being disjoint union of complete graphs, Chvátal and Harary [6] showed that any tree is 2K2-good, where tH denotes the union of t disjoint copies of graph H. Sudarsana, Adiwijaya and Musdalifah [15] proved that the n-vertex path Pn is 2Km-good for n ≥ 3 and m ≥ 2, and conjectured that any tree Tn with n vertices is 2Km-good. Recently, Pokrovskiy and Sudakov [11] proved that Pn is H-good for a graph with n ≥ 4v(H).…”