Graceful labeling

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

See also

References

  1. Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001. PostScript
  2. 1 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. 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. 1 2 Gallian, Joseph A. (1998), "A dynamic survey of graph labeling", Electronic Journal of Combinatorics 5: Dynamic Survey 6, 43 pp. (electronic), MR 1668059.
  6. 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.
  7. Fang, Wenjie (2010), A Computational Approach to the Graceful Tree Conjecture, arXiv:1003.3045. See also Graceful Tree Verification Project
  8. 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.
  9. Weisstein, Eric W., "Graceful graph", MathWorld.

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