Most graph labellings trace their origins to labellings presented by Alexander Rosa in his 1967 paper. Rosa identified three types of labellings, which he called , -, and -labellings. -labellings were later renamed as "graceful" by Solomon Golomb, and the name has been popular since.
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A graph colouring is a subclass of graph labellings. Vertex colourings assign different labels to adjacent vertices, while edge colourings assign different labels to adjacent edges.
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A closed loop is a cyclical path of adjacent vertices that never revisits the same vertex. Such a cycle can be thought of as the boundary of a hypothetical 2-cell. The -labellings of a graph that conserve momentum (i.e. which has zero boundary) up to redefinitions of (i.e.
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In mathematics, Ky Fan's lemma (KFL) is a combinatorial lemma about labellings of triangulations. It is a generalization of Tucker's lemma. It was proved by Ky Fan in 1952. In this example, where n = 2, there is no 2-dimensional alternating simplex (since the labels are only 1,2).
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There are a great many variations on the concept of magic labelling of a graph. There is much variation in terminology as well. The definitions here are perhaps the most common. Comprehensive references for magic labellings and magic graphs are Gallian (1998), Wallis (2001), and Marr and Wallis (2013).
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Fix a partition λ of n and a commutative ring k. The partition determines a Young diagram with n boxes. A Young tableau of shape λ is a way of labelling the boxes of this Young diagram by distinct numbers 1, \dots, n. A tabloid is an equivalence class of Young tableaux where two labellings are equivalent if one is obtained from the other by permuting the entries of each row.
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A sofic system is a subshift of finite type where different edges of the transition graph may correspond to the same symbol. It may be regarded as the set of labellings of paths through an automaton: a subshift of finite type then corresponds to an automaton which is deterministic.Pytheas Fogg (2002) p.205 A renewal system is defined to be the set of all infinite concatenations of some fixed finite collection of finite words.
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The hook length formula can be understood intuitively using the following heuristic, but incorrect, argument suggested by D. E. Knuth.. Given that each element of a tableau is the smallest in its hook and filling the tableau shape at random, the probability that cell (i,j) will contain the minimum element of the corresponding hook is the reciprocal of the hook length. Multiplying these probabilities over all i and j gives the formula. This argument is fallacious since the events are not independent. Knuth's argument is however correct for the enumeration of labellings on trees satisfying monotonicity properties analogous to those of a Young tableau.
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An even more conventional choice would be to label the vertices of P and Q by consecutive integers, but again there are two natural choices for how to align these labellings: Either Q_k is just clockwise from P_k or just counterclockwise. In most papers on the subject, some choice is made once and for all at the beginning of the paper and then the formulas are tuned to that choice. There is a perfectly natural way to label the vertices of the second iterate of the pentagram map by consecutive integers. For this reason, the second iterate of the pentagram map is more naturally considered as an iteration defined on labeled polygons.
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The ADE graphs and the extended (affine) ADE graphs can also be characterized in terms of labellings with certain properties, which can be stated in terms of the discrete Laplace operators or Cartan matrices. Proofs in terms of Cartan matrices may be found in . The affine ADE graphs are the only graphs that admit a positive labeling (labeling of the nodes by positive real numbers) with the following property: :Twice any label is the sum of the labels on adjacent vertices. That is, they are the only positive functions with eigenvalue 1 for the discrete Laplacian (sum of adjacent vertices minus value of vertex) – the positive solutions to the homogeneous equation: :\Delta \phi = \phi.
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