Graham's Tree Reconstruction Conjecture and a Waring-Type Problem on Partitions

Suppose G is a tree. Graham’s “Tree Reconstruction Conjecture” states that G is uniquely determined by the integer sequence |G|, |L(G)|, |L(L(G))|, |L(L(L(G)))|, . . ., where L(H) denotes the line graph of the graph H . Little is known about this question apart from a few simple observations. We show that the number of trees on n vertices which can be distinguished by their associated integer sequences is at least e 3/2). The proof strategy involves constructing a large collection of caterpillar graphs using partitions arising from the Prouhet-Tarry-Escott problem. We identify, but only partially resolve, an interesting question about representations of integers as sums of k powers of the parts of integer partitions.