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 , 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 , 6 ) = ( s + 1 ) ( n − 1 ) + 1 and for n ≥ 5 , s ≥ 1 , R ( T n , W s , 7 ) = ( s + 2 ) ( n − 1 ) + 1 . We also determine the exact value of R ( T n , W s , m ) for large n and s.