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 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
- 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 27 vertices are graceful; this result was shown by Aldred and McKay in 1998 using a computer program.[5][6] 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.[7]
- All wheel graphs, web graphs, Helm graphs, gear graphs, and rectangular grids are graceful.[5]
- All n-dimensional hypercubes are graceful.[8]
- 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.[9]
See also
References
- ↑ Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001. PostScript
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- ↑ Weisstein, Eric W., "Graceful graph", MathWorld.
Additional 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
