Ramsey Numbers of Trees Versus Multiple Copies of Books

Abstract

Given two graphs G and H, the Ramsey number R(G,H) is the minimum integer N such that any two-coloring of the edges of K_N 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 number of vertices in a minimum color class among all proper χ(G)-colorings of G. Burr showed that \(R(G,H) \ge (v(G) - 1)(\chi (H) - 1) + s(H)\) if G is connected and \(v(G) \ge s(H)\). A connected graph G is H-good if \(R(G,H) = (v(G) - 1)(\chi (H) - 1) + s(H)\). Let tH denote the disjoint union of t copies of graph H, and let \(G \vee H\) denote the join of G and H. Denote a complete graph on n vertices by K_n, and a tree on n vertices by T_n. Denote a book with n pages by B_n, i.e., the join \({K_2} \vee \overline {{K_n}} \). Erdős, Faudree, Rousseau and Schelp proved that T_n is B_m-good if \(n \ge 3m - 3\). In this paper, we obtain the exact Ramsey number of T_n versus 2B₂- Our result implies that T_n is 2B₂-good if n ≥ 5.