Every tree is a large subtree of a tree that decomposes Kn or Kn,n

Abstract

Let T be a tree with m edges. A well-known conjecture of Ringel states that T decomposes the complete graph $K_{2m+1}$. Graham and Häggkvist conjectured that T also decomposes the complete bipartite graph $K_{m,m}$. In this paper we show that there exists an integer n with n ≤[(3m - 1)/2] and a tree T₁ with n edges such that T₁ decomposes $K_{2n+1}$ and contains T. We also show that there exists an integer n' with n' ≥ 2m-1 and a tree T₂ with n' edges such that T₂ decomposes $K_{n',n'}$and contains T. In the latter case, we can improve the bound if there exists a prime p such that [3m/2] ≤ p < 2m - 1.