Graph labeling

When the edge labels are members of an ordered set (e.g., the real numbers), it may be called a weighted graph.

For example, the edges may be assigned weights representing the "cost" of traversing between the incident vertices.

[4] Rosa identified three types of labelings, which he called α-, β-, and ρ-labelings.

In his original paper, Rosa proved that all Eulerian graphs with size equivalent to 1 or 2 (mod 4) are not graceful.

Anton Kotzig himself has called the effort to prove the conjecture a "disease".

It is conjectured that trees are all harmonious if one vertex label is allowed to be reused.

Vertex colorings assign different labels to adjacent vertices, while edge colorings assign different labels to adjacent edges.