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.^[1] A graph which admits a graceful labeling is called a graceful graph.

The name "graceful labeling" is due to Solomon W. Golomb; this type of labeling was originally given the name β-labeling by Alexander Rosa in a 1967 paper on graph labelings.^[2]

A major conjecture in graph theory is the graceful tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, and sometimes abbreviated GTC.^[3] It hypothesizes that all trees are graceful. It is still an open conjecture, although a related but weaker conjecture known as "Ringel's conjecture" was partially proven in 2020.^[4][5][6] Kotzig once called the effort to prove the conjecture a "disease".^[7]

Another weaker version of graceful labelling is near-graceful labeling, in which the vertices can be labeled using some subset of the integers on [0, m + 1] such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints (this magnitude lies on [1, m + 1]).

Another conjecture in graph theory is Rosa's conjecture, named after Alexander Rosa, which says that all triangular cacti are graceful or nearly-graceful.^[8]

A graceful graph with edges 0 to m is conjectured to have no fewer than ${\displaystyle \left\lceil {\sqrt {3m+{\tfrac {9}{4))))\right\rfloor }$ vertices, due to sparse ruler results. This conjecture has been verified for all graphs with 213 or fewer edges.

Selected results

• In his original paper, Rosa proved that an Eulerian graph with number of edges m ≡ 1 (mod 4) or m ≡ 2 (mod 4) cannot be graceful.^[2]
• Also in his original paper, Rosa proved that the cycle C_n is graceful if and only if n ≡ 0 (mod 4) or n ≡ 3 (mod 4).
• All path graphs and caterpillar graphs are graceful.^[2]
• All lobster graphs with a perfect matching are graceful.^[9]
• All trees with at most 27 vertices are graceful; this result was shown by Aldred and McKay in 1998 using a computer program.^[10][11] This was extended to trees with at most 29 vertices in the Honours thesis of Michael Horton.^[12] Another extension of this result up to trees with 35 vertices was claimed in 2010 by the Graceful Tree Verification Project, a distributed computing project led by Wenjie Fang.^[13]
• All wheel graphs, web graphs, helm graphs, gear graphs, and rectangular grids are graceful.^[10]
• All n-dimensional hypercubes are graceful.^[14]
• All simple connected graphs with four or fewer vertices are graceful. The only non-graceful simple connected graphs with five vertices are the 5-cycle (pentagon); the complete graph K₅; and the butterfly graph.^[15]