A graceful graph is a graph that can be gracefully labeled. Special cases of graceful graphs include the utility graph (Gardner 1983) and Petersen graph. A graph that cannot be gracefully labeled is called an ungraceful (or sometimes disgraceful) graph.

Graceful graphs may be connected or disconnected; for example, the graph disjoint union of the singleton graph and a complete graph is graceful iff (Gallian 2018).

Although an unpublished result of Erdős says that most graphs are not graceful (Graham and Sloane 1980), most graphs that have some sort of regularity of structure are graceful (Gallian 2018).

It is an unsolved and apparently very difficult problem to determine which graphs are graceful. One reason for its difficulty is that a subgraph of a graceful graph need not be graceful (Seoud and Wilson 1993).

In order for a graph to be graceful, it must be without loops or multiple edges. A graph on vertices and edges must also satisfy

in order to be graceful, since otherwise there are not enough integers less than or equal to m to cover all the vertices. Another criterion than can be used to determine a graph is ungraceful is due to Rosa (1967), who proved that an Eulerian graph with edge count congruent to 1 or 2 (mod 4) is ungraceful.

The numbers of graceful graphs on , 2, ... nodes are 1, 1, 2, 7, 22, 126, ... (OEIS A308548), while the corresponding numbers of connected graceful graphs are 1, 1, 2, 6, 18, 106, ... (OEIS A308549). The numbers of ungraceful graphs on , 2, ... nodes are 0, 1, 2, 4, 12, 30, 85, ... (OEIS A308556), with the corresponding numbers of connected ungraceful graphs 0, 0, 0, 0, 3, 6, 34, ... (OEIS A308557), the first few of which are illustrated above.

A graph that contains a single fundamentally distinct graceful labeling (i.e., a unique labeling modulo the graph automorphism group and with respect to subtractive complementation) may be termed a uniquely graceful graph, and a graph possessing the maximum possible number of fundamentally distinct labelings (possibly subject to some additional condition such as possessing no isolated vertices) may be termed a maximally graceful graph.

Parametrized families of graceful graphs include the following:

1. antiprism graphs,

2. banana trees,

3. book graphs ,

4. Cameron graphs,

5. caterpillar graphs,

6. complete graphs iff (Golomb 1974),

7. complete bipartite graphs (Golomb 1974),

8. crossed prism graphs,

9. cycle graphs iff ,

10. firecracker graphs,

11. gear graphs,

12. grid graphs ,

13. helm graphs,

14. hypercube graphs ,

15. ladder graphs ,

16. Lindgren-Sousselier graphs,

17. Möbius ladders ,

18. Mongolian tent graphs,

19. pan graphs,

20. path graphs ,

21. Platonic graphs (Gardner 1983, pp. 158 and 163-164),

22. prism graphs ,

23. star graphs ,

24. sunlet graphs ,

25. tadpole graphs,

26. web graphs, and

27. wheel graphs (Frucht 1988).

The -barbell graph is ungraceful from up to at least (E. Weisstein, Sep. 19, 2025) and likely for all larger .

In 1965, Kotzig conjectured that all trees are graceful, an almost certainly true conjecture known as the graceful tree theorem that remains unproven to this day.

It has also been conjectured that all unicyclic graphs except the cycle graph with or 2 (mod 4) are graceful (Truszczyński 1984, Gallian 2018).

See also

Edge-Graceful Graph, Graceful Labeling, Graceful Permutation, Graceful Pi-Way, Graceful Tree Theorem, Harmonious Graph, Labeled Graph, Maximally Graceful Graph, Ungraceful Graph, Uniquely Graceful Graph

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References

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Weisstein, Eric W. "Graceful Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GracefulGraph.html

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