Harmonious Labeling
A graph G with e edges and v vertices is harmonious if we can label the vertices {0, . . . , v ā1}, and label the edges the sum of the neighboring vertex labels mod e, so that we use {0, . . . , eā1} (see fig. 3).
By trying to find a way to build a harmonious graph which could be continued indefinitely, we came upon the type of graph shown in fig. 4.
Theorem. All Dn are harmonious.
Each Dn can be built from the graph Dn-1 by adding two vertices and four edges (see fig. 4). If Dn is built in this way, it will be harmonious at every stage, so the final graph Dn will be harmonious.
An infinite graph G is harmonious if we can label the vertices {0, . . . ,∞}, and label the edges the sum of the neighboring vertex labels so that we use {1, . . . ,∞}.
The infinite graph D is the union of all Dn for all n.
Since the method used in the proof that all Dn are harmonious can be repeated indefinitely, D is also harmonious.