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 between 0 and m inclusive, such that no two vertices share a label, and such that each edge is uniquely identified by the positive, or absolute difference between its endpoints. A graph which admits a graceful labeling is called a graceful graph.

The name "graceful labeling" is due to Solomon W. Golomb; this class of labelings was originally given the name β-labelings by Alexander Rosa in a 1967 paper on graph labelings.

A major unproven conjecture in graph theory is the Graceful Tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, which hypothesizes that all trees are graceful. The Ringel-Kotzig conjecture is also known as the "graceful labeling conjecture". Kotzig once called the effort to prove the conjecture a "disease".

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