Frequency partition of a graph

In graph theory, a discipline within mathematics, the frequency partition of a graph (simple graph) is a partition of its vertices grouped by their degree. For example, the degree sequence 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 bipartite graph 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.

A graph with frequency partition 6 = 3 + 2 + 1.
A bipartite graph with frequency partition 9 = 5 + 3 + 1.
A graph with frequency partition 7 = 6 + 1.

In general, there are many non-isomorphic graphs with a given frequency partition. A graph and its complement have the same frequency partition. For any partition p = f₁ + f₂ + ... + f_k of an integer p > 1, other than p = 1 + 1 + 1 + ... + 1, there is at least one (connected) simple graph having this partition as its frequency partition.^[1]

Frequency partitions of various graph families are completely identifieds; frequency partitions of many families of graphs are not identified.

Frequency partitions of Eulerian graphs

For a frequency partition p = f₁ + f₂ + ... + f_k of an integer p > 1, its graphic degree sequence is denoted as ((d₁)^f₁,(d₂)^f₂, (d₃)^f₃, ..., (d_k) ^f_k) where degrees d_i's are different and f_i ≥ f_j for i < j. Bhat-Nayak et al. (1979) showed that a partition of p with k parts, k ≤ integral part of $(p-1)/2$ is a frequency partition^[2] of a Eulerian graph and conversely.

Frequency partition of trees, Hamiltonian graphs, tournaments and hypegraphs

The frequency partitions of families of graphs such as trees,^[3] Hamiltonian graphs^[4] directed graphs and tournaments^[5] and to k-uniform hypergraphs.^[6] have been characterized.

Unsolved problems in frequency partitions

The frequency partitions of the following families of graphs have not yet been characterized:

References

↑ Chinn, P. Z. (1971), "The frequency partition of a graph. Recent Trends in Graph Theory", Lecture Notes in Mathematics, Berlin: Springer-Verlag, 186, pp. 69–70
↑ Rao, Siddani Bhaskara; Bhat-Nayak, Vasanti N.; Naik, Ranjan N. (1979), "Characterization of frequency partitions of Eulerian graphs", Proceedings of the Symposium on Graph Theory (Indian Statist. Inst., Calcutta, 1976), ISI Lecture Notes, 4, Macmillan of India, New Delhi, pp. 124–137, MR 553937 . Also in Lecture Notes in Mathematics, Combinatorics and Graph Theory, Springer-Verlag, New York, Vol. 885 (1980), p 500.
↑ Rao, T. M. (1974), "Frequency sequences in Graphs", Journal of Combinatorial Theory B, 17: 19–21, doi:10.1016/0095-8956(74)90042-2
↑
- Bhat-Nayak, Vasanti N.; Naik, Ranjan N. & Rao, S. B. (1977), "Frequency partitions: forcibly pancyclic and forcibly nonhamiltonian degree sequences", Discrete Mathematics, 20: 93–102, doi:10.1016/0012-365x(77)90049-8
↑ Alspach, B. & Reid, K. B. (1978), "Degree Frequencies in Digraphs and Tournaments", Journal of Graph Theory, 2: 241–249, doi:10.1002/jgt.3190020307
↑ Bhat-Nayak, V. N. & Naik, R. N. (1985), "Frequency partitions of k-uniform hypergraphs", Utilitas Math., 28: 99–104
↑ S. B. Rao, 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

External section

Berge, C. (1989), Hypergraphs, Combinatorics of Finite sets, Amsterdam: North-Holland, ISBN 0-444-87489-5

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