Perfect distance forests

In 1975, Leech introduced the problem of finding trees whose edges can be labeled with positive integers so that the sums of the edge labels between vertices is {1, 2, · · · , (n 2 )}, where n is the number of vertices. We refer to such trees as perfect distance trees. In a previous paper, this problem was generalized to graphs in general and in particular, forests. Here, we show that tK1,2 can be labeled as a perfect distance forest if and only if t ∼= 0, 1(mod 4) and that tK1,3 can always be labeled as a perfect distance forest. Examples of labelings for tP3 and sK1,3 ∪ tP3 as perfect distance forests are given. The most important result on perfect distance graphs, Taylor’s Theorem, is used to rule out the existence of many forests with same size components. Finally, we examine the case of perfect distance forests with 2 components. For n ≤ 16, we determine all pairs n1, n2 such that there is a perfect distance forest with components of size n1 and n2.