Modular Leech Trees of Order at Most 8
In 1975, John Leech asked when can the edges of a tree on vertices be labeled with positive integers such that the sums along the paths are exactly the integers . 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 where and examined the cases for . In this paper we show that no modular Leech trees of order 7 exist, and we find all modular Leech trees of order 8.
A tree on vertices is said to be a Leech tree if its edges can be weighted with positive integers in such a way that each of the paths has a distinct weight from the set . The weight of a path is found by summing all of its edge weights. Leech  found the five examples shown in Figure 1, which are to date the only ones known. In 1977, Taylor  proved that, in order for a Leech tree of order to exist, it must be that or for some integer . Since then it has been shown by several authors [3–5] that no Leech trees of order , , or exist, leaving as the smallest open case. Székely et al.  have conjectured that no additional Leech trees exist. Since Leech trees are so difficult to come by, we consider the generalization to modular Leech trees.
Let be a tree on vertices and let . We say that is a modular Leech tree if there exists an edge weighting function such that each of the paths within has a distinct weight from with the sums taken modulo . We call such an edge weighting function a -Leech labeling. Since the pathweights are all distinct, the function induces a bijection between the paths of and the elements of the group . We use to refer to this bijection as well.
Note that a “normal” Leech tree of order is also a modular Leech tree over in which none of the path sums, before applying the mod operation, have a weight greater than . Thus Leech’s original five examples provide us with five modular Leech trees. The only other example previously known is the tree of order found in  shown in Figure 2. It was also shown in  that no modular Leech tree of order 5 exists.
2. Taylor’s Condition for Modular Leech Trees
For the normal Leech trees, Taylor’s condition restricts the possible orders severely. However, with modular Leech trees over , Taylor’s condition only applies when is even. The proof is very similar to half of Taylor’s proof.
Theorem 1. Suppose that is a modular Leech tree of order and or 3 ( 4); then for some integer .
Proof. Assume that is a modular Leech tree of order and that or 3 (mod 4). Since or 3 (mod 4), we have that is odd and thus the modulus is even.
We color each vertex of the tree black or white as follows: start at any vertex and color it black. From we color all other vertices by traversing the edges of the graph. We keep the same color across edges with even weight and change colors across edges with odd weight. When all vertices are colored, an edge connects different colored vertices if and only if its weight is odd. Furthermore, since the modulus is even, for any vertices , the path from to has odd weight if and only if and are colored with opposite colors.
Now we count the number of odd paths in two ways: let and be the number of black and white vertices, respectively. Thus the number of odd paths is . Also, since there are paths and is odd, the number of odd paths is . Putting these together gives . Substituting for on the right side leads to and the theorem is proved.
The proof of Theorem 1 makes use of the fact that, under the assumptions of the theorem, a path has odd weight if and only if it contains an odd number of odd-weight edges. This fact does not hold when the modulus is odd. (For example, consider a path on four vertices with edges weighted 3, 5, and 1 and examine the path-weights mod 7.) Recall that a normal Leech tree on vertices is also a modular Leech tree over where and by Taylor’s condition or for some integer . We can conclude the following.
Theorem 2. If there exists a modular Leech tree of order , and is not for some integer , then or 1 ( 4).
3. Computational Results
We have already seen modular Leech trees of orders 2, 3, 4, and 6. By Leach and Walsh  and Theorem 2 we know that none exist for 5 or 7. To examine larger values of , we use computer search. The following theorem and corollary reduce the space that must be searched.
Theorem 3. Let be an integer and . Suppose that is a -Leech labeling of a tree . Then for any satisfying , the function defined by is also a -Leech labeling of .
Proof. Let be the set of all paths in . We can consider as a bijection between and , so for every nonzero , there exists a path with weight . Now define by .
We now show that is bijective: let and be paths in and suppose that . Then . Since , exists and thus . Since is a bijective, and thus is also bijective. Since is bijective, is a Leech labeling of .
Corollary 4. If there exists a -Leech labeling of a tree and , then there exists a -Leech labeling of in which edge has weight .
For , there are 23 distinct unlabeled trees. By computer search, we find that there is one modular Leech tree for and the edge-weighting function is unique, up to group and graph isomorphism. It is shown in Figure 3.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.