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

A graceful labeling. Vertex labels are in black, edge labels in red.

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.

A graceful graph with 27 edges and 9 vertices

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]

See also

• Edge-graceful labeling
• List of conjectures

References

1. Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001. PostScript
2. Rosa, A. (1967), "On certain valuations of the vertices of a graph", Theory of Graphs (Internat. Sympos., Rome, 1966), New York: Gordon and Breach, pp. 349–355, MR 0223271.
3. Wang, Tao-Ming; Yang, Cheng-Chang; Hsu, Lih-Hsing; Cheng, Eddie (2015), "Infinitely many equivalent versions of the graceful tree conjecture", Applicable Analysis and Discrete Mathematics, 9 (1): 1–12, doi:10.2298/AADM141009017W, MR 3362693
4. Montgomery, Richard; Pokrovskiy, Alexey; Sudakov, Benny (2020). "A proof of Ringel's Conjecture". arXiv:2001.02665 [math.CO].
5. Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 0668845.
6. Hartnett, Kevin. "Rainbow Proof Shows Graphs Have Uniform Parts". Quanta Magazine. Retrieved 2020-02-29.
7. Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 0668845.
8. Rosa, A. (1988), "Cyclic Steiner Triple Systems and Labelings of Triangular Cacti", Scientia, 1: 87–95.
9. Morgan, David (2008), "All lobsters with perfect matchings are graceful", Bulletin of the Institute of Combinatorics and Its Applications, 53: 82–85, hdl:10402/era.26923.
10. Gallian, Joseph A. (1998), "A dynamic survey of graph labeling", Electronic Journal of Combinatorics, 5: Dynamic Survey 6, 43 pp. (389 pp. in 18th ed.) (electronic), MR 1668059.
11. Aldred, R. E. L.; McKay, Brendan D. (1998), "Graceful and harmonious labellings of trees", Bulletin of the Institute of Combinatorics and Its Applications, 23: 69–72, MR 1621760.
12. Horton, Michael P. (2003), Graceful Trees: Statistics and Algorithms.
13. Fang, Wenjie (2010), A Computational Approach to the Graceful Tree Conjecture, arXiv:1003.3045, Bibcode:2010arXiv1003.3045F. See also
14. Kotzig, Anton (1981), "Decompositions of complete graphs into isomorphic cubes", Journal of Combinatorial Theory, Series B, 31 (3): 292–296, doi:10.1016/0095-8956(81)90031-9, MR 0638285.
15. Weisstein, Eric W. "Graceful graph". MathWorld.

External links

• Numberphile video about graceful tree conjecture
• Graceful labeling in mathworld

Further reading

• (K. Eshghi) Introduction to Graceful Graphs, Sharif University of Technology, 2002.
• (U. N. Deshmukh and Vasanti N. Bhat-Nayak), New families of graceful banana trees – Proceedings Mathematical Sciences, 1996 – Springer
• (M. Haviar, M. Ivaska), Vertex Labellings of Simple Graphs, Research and Exposition in Mathematics, Volume 34, 2015.
• (Ping Zhang), A Kaleidoscopic View of Graph Colorings, SpringerBriefs in Mathematics, 2016 – Springer