Some arrowing results for trees versus complete graphs

For an arbitrary tree T of order m and an arbitrary positive integer n , Chvátal proved that the Ramsey number r(T, K n ) = 1 + ( m − 1) ( n − 1). for graphs G, G 1 , and G 2 , we say that G arrows G 1 and G 2 , written G → ( G 1 , G 2 ), if for every factorization G = R ⊕ B , either G 1 is a subgraph of R or G 2 is a subgraph of B . it is shown that (i) for each l ≥ 2, K 1 + (m−1)(n−1) − E(K 1 ) → ( T, K n ) for m ≥ 2/ − 1 and n ≥ 2; (ii) K 1 +,(m −1)(n −1) − E(H) → ( T, K n ), where H is any tree of order m − 1, m ≥ 3 and n ≥ 2. It is further shown that result (i) is sharp with respect to the inequality m ≥ 2/ − 1; in particular, examples are given to show that K 1 + (2l−3)( n −1) E(K 1 ) ↛ ( P 21−2 , K n ) for all n ≥ 2, where P 21−2 denotes the path of order 21 − 2. Also result (ii) is sharp with respect to the order of H ; examples aregiven to show that K 1 + (m−1)(n−1) − E(K( 1, m − 1)) ↛( T, K n ) for any tree T of order m and any n ≥ 2.