Every tree is a large subtree of a tree that decomposes or

Let T be a tree with m edges. A well-known conjecture of Ringel states that T decomposes the complete graph K 2 m + 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 ≤ ⌈ ( 3 m − 1 ) / 2 ⌉ and a tree T 1 with n edges such that T 1 decomposes K 2 n + 1 and contains T . We also show that there exists an integer n ′ with n ′ ≥ 2 m − 1 and a tree T 2 with n ′ edges such that T 2 decomposes K n ′ , n ′ and contains T . In the latter case, we can improve the bound if there exists a prime p such that ⌈ 3 m / 2 ⌉ ≤ p < 2 m − 1 .