A large tree is tKm-good

Abstract

Given two graphs G and H, the Ramsey number r(G,H) is the minimum integer N such that any red-blue coloring of the edges of KN contains either a red copy of G or a blue copy of H. Let v(G) denote the number of vertices of G, and χ(G) denote 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 H with v(G)≥s(H), r(G,H)≥(v(G)−1)(χ(H)−1)+s(H). A connected graph G is called H-good if r(G,H)=(v(G)−1)(χ(H)−1)+s(H). Chvátal [7] showed that any tree is Km-good for m≥2, where Km denotes a complete graph with m vertices. Let tKm denote t vertex disjoint copies of Km. Recently, Hu and Peng [13] proved that any tree Tn is 2Km-good for n≥3 and m≥2, they [14] also showed that any large tree is tK2-good for t≥3. In this paper, we show that any large tree is tKm-good for t≥3 and m≥3.