On the Ramsey numbers for the tree graphs versus certain generalised wheel graphs

Given two simple graphs G and H, the Ramsey number R(G, H) is the smallest integer n such that for any graph of order n, either it contains G or its complement contains H. Let T-n be a tree graph of order n and W-s,W-m be the generalised wheel graph K-s + C-m. In this paper, we show that for n >= 5, s >= 2, R(T-n, W-s,W- 6) = (s + 1)(n - 1) + 1 and for n >= 5, s >= 1, R(T-n, W-s,W-7) = (s + 2)(n - 1) + 1. We also determine the exact value of R(T-n, W-s,W-m) for large nand s. (C) 2021 Elsevier B.V. All rights reserved.