Modular Leech Trees of Order at Most 8

Abstract

In 1975, John Leech asked when can the edges of a tree on n vertices be labeled with positive integers such that the sums along the paths are exactly the integers 1, 2, …, n2. He found five such trees, and no additional trees have been discovered since. In 2011 Leach and Walsh introduced the idea of labeling trees with elements of the group Zk where k=n2+1 and examined the cases for n≤6. In this paper we show that no modular Leech trees of order 7 exist, and we find all modular Leech trees of order 8.