Decomposing almost complete graphs by random trees

An old conjecture of Ringel states that every tree with m edges decomposes the complete graph K2m+1. The best lower bound for the order of a complete graph decomposed by a given tree with m edge is O(m). We show that asymptotically almost surely a random tree with m edges and p = 2m + 1 a prime decomposes K2m+1(r) for every r ≥ 2, the graph obtained from the complete graph K2m+1 by replacing each vertex by a coclique of order r. As a consequence of the main result we obtain approximations to Ringel’s conjecture for random trees of almost complete graphs of linear order with the size of the tree. MSC2010: 05C51, 05C80