92:
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of the left-hand graph below is (3, 3, 3, 2, 2, 1) and its frequency partition is 6 = 3 + 2 + 1. This indicates that it has 3 vertices with some degree, 2 vertices with some other degree, and 1 vertex with a third degree. The degree sequence of the
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in the middle below is (3, 2, 2, 2, 2, 2, 1, 1, 1) and its frequency partition is 9 = 5 + 3 + 1. The degree sequence of the right-hand graph below is (3, 3, 3, 3, 3, 3, 2) and its frequency partition is 7 = 6 + 1.
510:, A survey of the theory of potentially p-graphic and forcibly p-graphic sequences, in: S. B. Rao edited., Combinatorics and Graph Theory Lecture Notes in Math., Vol. 885 (Springer, Berlin, 1981), 417-440
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Frequency partitions of various graph families are completely identified; frequency partitions of many families of graphs are not identified.
361:
Rao, Siddani
Bhaskara; Bhat-Nayak, Vasanti N.; Naik, Ranjan N. (1979), "Characterization of frequency partitions of Eulerian graphs",
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141:= 1 + 1 + 1 + ... + 1, there is at least one (connected) simple graph having this partition as its frequency partition.
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In general, there are many non-isomorphic graphs with a given frequency partition. A graph and its
426:(1977), "Frequency partitions: forcibly pancyclic and forcibly nonhamiltonian degree sequences",
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The frequency partitions of the following families of graphs have not yet been characterized:
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Bhat-Nayak, V. N. & Naik, R. N. (1985), "Frequency partitions of k-uniform hypergraphs",
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350:, Lecture Notes in Mathematics, vol. 186, Berlin: Springer-Verlag, pp. 69–70
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Proceedings of the
Symposium on Graph Theory (Indian Statist. Inst., Calcutta, 1976)
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365:, ISI Lecture Notes, vol. 4, Macmillan of India, New Delhi, pp. 124–137,
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Frequency partition of trees, Hamiltonian graphs, tournaments and hypegraphs
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374:. Also in Lecture Notes in Mathematics, Combinatorics and Graph Theory,
236:(1979) showed that a partition of p with k parts, k ≤ integral part of
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The frequency partition of a graph. Recent Trends in Graph Theory
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is a frequency partition of a
Eulerian graph and conversely.
463:(1978), "Degree Frequencies in Digraphs and Tournaments",
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A bipartite graph with frequency partition 9 = 5 + 3 + 1.
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The frequency partitions of families of graphs such as
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have the same frequency partition. For any partition
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Rao, T. M. (1974), "Frequency sequences in Graphs",
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73:A graph with frequency partition 6 = 3 + 2 + 1.
8:
97:A graph with frequency partition 7 = 6 + 1.
51:grouped by their degree. For example, the
529:Hypergraphs, Combinatorics of Finite sets
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391:Journal of Combinatorial Theory, Series B
307:Unsolved problems in frequency partitions
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148:Frequency partitions of Eulerian graphs
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378:, New York, Vol. 885 (1980), p 500.
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531:, Amsterdam: North-Holland,
443:10.1016/0012-365x(77)90049-8
405:10.1016/0095-8956(74)90042-2
303:. have been characterized.
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422:; Naik, Ranjan N. &
465:Journal of Graph Theory
269:{\displaystyle (p-1)/2}
188:graphic degree sequence
477:10.1002/jgt.3190020307
420:Bhat-Nayak, Vasanti N.
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429:Discrete Mathematics
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37:frequency partition
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436:: 93–102,
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330:References
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