Graceful labeling

In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with some subset of the integers from 0 to m inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and m inclusive.

[2] A major open problem in graph theory is the graceful tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, and sometimes abbreviated GTC (not to be confused with Kotzig's conjecture on regularly path connected graphs).

[4][5][6] Kotzig once called the effort to prove the conjecture a "disease".

Another conjecture in graph theory is Rosa's conjecture, named after Alexander Rosa, which says that all triangular cacti are graceful or nearly-graceful.

vertices, due to sparse ruler results.