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

Abstract

Given two graphs G and H, the Ramsey number R(G, H) is the minimum integer N such that any coloring of the edges of KN in red or blue yields a red G or a blue H. Let v(G) be the number of vertices of G and χ(G) be the chromatic number of G. Let s(G) denote the chromatic surplus of G, the cardinality of a minimum color class taken over all proper colorings of G with χ(G) colors. Burr [3] showed that for a connected graph G and a graph Chvátal [5] showed that any tree is Km-good for m ≥ 2, where Km denotes a complete graph with m vertices. 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). Balla, Pokrovskiy and Sudakov [2] showed that for all ∆ and k, there exists a constant C ∆,k such that for any tree T with maximum degree at most ∆ and any H withIn this paper, we explore the Ramsey number of forest versus disjoint union of complete graphs. We first confirm the conjecture by Sudarsana, Adiwijaya and Musdalifah [15] that any tree is 2Kmgood for n ≥ 3 and m ≥ 2. A key proposition in our proof is that a tree with n vertices can be obtained from any tree with n vertices by performing a series of "Stretching" and "Expanding" operations. On this foundation, we show that these two operations preserve the "2Km-goodness" property, and confirm that any tree is 2Km-good. We also prove a conclusion which yields that Tn is Km ∪ K l -good, where Km ∪ K l is the disjoint union of Km and K l , m > l ≥ 2. Furthermore, we extend the Ramsey goodness of connected graphs to disconnected graphs and study the relation between the Ramsey number of the components of a disconnected graph F versus a graph H. We show that if each component of a graph F is H-good, then F is H-good. Our result implies the exact value of R(F, Km ∪ K l ), where F is a forest and m, l ≥ 2. It's interesting to explore what kind of graph F can satisfy that F is H-good and G-good implies that F is H ∪ G-good when the number of vertices in F is sufficiently large.