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 $$K_N$$ in red or blue yields a red G or a blue H. Let $$\chi (G)$$ be the chromatic number of G, and s(G) denote the chromatic surplus of G, the cardinality of a minimum color class taken over all proper colorings of G with $$\chi (G)$$ colors. A connected graph G is called H-good if $$R(G,H)=(v(G)-1)(\chi (H)-1)+s(H)$$ . Chvátal (J. Graph Theory 1:93, 1977) showed that any tree is $$K_m$$ -good for $$m\ge 2$$ , where $$K_m$$ denotes a complete graph with m vertices. Let tH denote the union of t disjoint copies of graph H. Sudarsana et al. (Comput. Sci. 6196, Springer, Berlin, 2010) proved that the n-vertex path $$P_n$$ is $$2K_m$$ -good for $$n\ge 3$$ and $$m\ge 2$$ , and conjectured that any n-vertex tree $$T_n$$ is $$2K_m$$ -good. In this paper, we confirm this conjecture and prove that $$T_n$$ is $$2K_m$$ -good for $$n\ge 3$$ and $$m\ge 2$$ . We also prove a conclusion which yields that $$T_n$$ is $$(K_m\cup K_l)$$ -good, where $$K_m\cup K_l$$ is the disjoint union of $$K_m$$ and $$K_l$$ , $$m>l\ge 2$$ . 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 $$\textrm{F}$$ 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 $$R(F,K_m\cup K_l)$$ , where F is a forest and $$m,l\ge 2$$ .