Trees are almost prime

Let S-n denote the graph on {1,..., n) in which two numbers are adjacent if and only if they are coprime. Around 1980 Entringer conjectured that Sn contains every tree of order n as a subgraph. Here we show that this conjecture is true for all n <= 50. Further positive evidence is provided by our main result that S, contains every tree of order (1 - o(1))n. (c) 2006 Published by Elsevier B.V.