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

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