Modular gracious labellings of trees

A gracious labelling g of a tree is a graceful labelling in which, treating the tree as a bipartite graph, the label of any edge (d,u) (d a 'down' and u an 'up' vertex) is g(u) - g(d). A gracious k-labelling is one such that each residue class modulo k has teh 'correct' numbers of vertex and edge labels -- that is, the numbers that arise by interpreting the labels of a gracious labelling modulo k. In this paper it is shown that every non-null tree has a gracious k-labelling for each k = 2,3,4,5