Graceful labeling

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

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.[1] 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.[2]

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".[3]

Selected results[edit]

  • In his original paper, Rosa proved that an Eulerian graph with number of edges m ≡ 1 (mod 4) or m ≡ 2 (mod 4) can't be graceful.[2]
  • Also in his original paper, Rosa proved that the cycle Cn is graceful if and only if n ≡ 0 (mod 4) or n ≡ 3 (mod 4).
  • All path graphs and caterpillar graphs are graceful.
  • All lobster graphs with a perfect matching are graceful.[4]
  • All trees with at most 29 vertices are graceful; this result was shown by Michael Horton in his Honours thesis in 2003.[5]
  • All trees with at most 27 vertices are graceful; this result was shown by Aldred and McKay in 1998 using a computer program.[6][7] An extension of this (using a different computational method) up to trees with 35 vertices was claimed in 2010 by the Graceful Tree Verification Project, a distributed computing project led by Wenjie Fang.[8]
  • All wheel graphs, web graphs, helm graphs, gear graphs, and rectangular grids are graceful.[6]
  • All n-dimensional hypercubes are graceful.[9]
  • All simple graphs with four or fewer vertices are graceful. The only non-graceful simple graphs with five vertices are the 5-cycle (pentagon); the complete graph K5; and the butterfly graph.[10]

See also[edit]


  1. ^ Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001. PostScript
  2. ^ a b 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. ^ Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 668845 .
  4. ^ Morgan, David (2008), "All lobsters with perfect matchings are graceful", Bulletin of the Institute of Combinatorics and its Applications, 53: 82–85 .
  5. ^ Horton, Michael P. (2003), Graceful Trees: Statistics and Algorithms (PDF) .
  6. ^ a b 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 .
  7. ^ 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 .
  8. ^ Fang, Wenjie (2010), A Computational Approach to the Graceful Tree Conjecture, arXiv:1003.3045free to read . See also Graceful Tree Verification Project
  9. ^ 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 638285 .
  10. ^ Weisstein, Eric W. "Graceful graph". MathWorld. 

Additional reading[edit]

  • (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