The Ramsey numbers for disjoint unions of trees

Abstract

For given graphs G and $H,$ the Ramsey number $R(G,H)$ is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains $H.$ In this paper, we investigate the Ramsey number $R(\cup G,H)$, where G is a tree and H is a wheel $W_m$ or a complete graph $K_m$. We show that if $n⩾3$, then $R({\mathit{kS}}_n,W_4)=(k+1)n$ for $k⩾2$, even n and $R({\mathit{kS}}_n,W_4)=(k+1)n-1$ for $k⩾1$ and odd n. We also show that $R({\bigcup }_{i=1}^k{T}_{n_i},K_m)=R(T_{n_k},K_m)+{\sum }_{i=1}^{k-1}{n}_i$.