Graceful and harmonious labellings of trees

We establish that all trees on at most 27 vertices admit graceful labellings and all trees on at most 26 vertices admit harmonious labellings. A graceful labelling of a graph G with q edges is an injection f : V (G) → {0, 1, 2, . . . , q} such that when each edge xy ∈ E(G) is assigned the label, |f(x) − f(y)|, all of the edge labels are distinct. A graph which admits a graceful labelling is said to be graceful. This idea was introduced by Rosa [5] where it was shown that if all trees are graceful, then the Ringel-Kotzig conjecture is true. (Ringel [4] conjectured that K2n+1 can be decomposed into 2n + 1 subgraphs which are isomorphic to a given tree with n edges. Kotzig conjectured K2n+1 can be cyclically decomposed into 2n + 1 subgraphs which are isomorphic to a given tree with n edges.) In the same paper, Rosa showed that several families of trees are graceful and also that all trees on at most 16 vertices are graceful. Since this paper many papers have been written about graceful graphs and in particular graceful trees (see [1],[2]) but, apart from several more families, there has been no advance from 16 on