On the Ramsey numbers of non-star trees versus connected graphs of order six

This paper completes our studies on the Ramsey number r(Tn, G) for trees Tn of order n and connected graphs G of order six. If χ(G) ≥ 4, then the values of r(Tn, G) are already known for any tree Tn. Moreover, r(Sn, G), where Sn denotes the star of order n, has been investigated in case of χ(G) ≤ 3. If χ(G) = 3 and G 6= K2,2,2, then r(Sn, G) has been determined except for some G and some small n. Partial results have been obtained for r(Sn,K2,2,2) and for r(Sn, G) with χ(G) = 2. In the present paper we investigate r(Tn, G) for non-star trees Tn and χ(G) ≤ 3. Especially, r(Tn, G) is completely evaluated for any non-star tree Tn if χ(G) = 3 where G 6= K2,2,2, and r(Tn,K2,2,2) is determined for a class of trees Tn with small maximum degree. In case of χ(G) = 2, r(Tn, G) is investigated for Tn = Pn, the path of order n, and for Tn = B2,n−2, the special broom of order n obtained by identifying the centre of a star S3 with an end-vertex of a path Pn−2. Furthermore, the values of r(B2,n−2, Sm) are determined for all n and m with n ≥ m− 1. As a consequence of this paper, r(F,G) is known for all trees F of order at most five and all connected graphs G of order at most six.