Locally Harmonious Labeling
A graph G with e edges and v vertices is locally harmonious if we can label the vertices {0, . . . , v − 1}, and the edge labels around each vertex represent its degree (fig. 5).
We started our investigation of locally harmonious labelings with the path.
Theorem. All paths are locally harmonious.
In fig. 6, an arbitrarily long, locally harmonious path is built by adding each new vertex to one of the ends in a pattern that results in alternating pairs of even and odd vertex labels.
An infinite graph is locally harmonious if we can label the vertices {0, . . . ,∞}, and the edge labels represent the number of edges it has.
Theorem. Assume G is a finite locally harmonious graph with at least one vertex of degree one. There exists an infinite graph G' ∋ G that is locally harmonious.
A semi-infinite path can be connected to the vertex of degree one. Then, it can be labeled with alternating pairs of even and odd labels. The resulting path is infinite and locally harmonious.