RAMSEY NUMBERS FOR TREES

Abstract

AbstractForn≥5, letT′ndenote the unique tree onnvertices with Δ(T′n)=n−2, and letT*n=(V,E) be the tree onnvertices withV={v0,v1,…,vn−1} andE={v0v1,…,v0vn−3,vn−3vn−2,vn−2vn−1}. In this paper, we evaluate the Ramsey numbersr(Gm,T′n) andr(Gm,T*n) , whereGmis a connected graph of orderm. As examples, forn≥8 we haver(T′n,T*n)=r(T*n,T*n)=2n−5 , forn>m≥7 we haver(K1,m−1,T*n)=m+n−3 orm+n−4 according to whetherm−1∣n−3 orm−1∤n−3 , and form≥7 andn≥(m−3)2+2 we haver(T*m,T*n)=m+n−3 orm+n−4 according to whetherm−1∣n−3 orm−1∤n−3 .