Frequency partition of a graph

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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.

  • Nonplanar no subgraph K 3 3.svg

    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 + ... + fk 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 + ... + fk of an integer p > 1, its graphic degree sequence is denoted as ((d1)f1,(d2)f2, (d3)f3, ..., (dk) fk) where degrees di's are different and fi ≥ fj 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

  1. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  2. ↑ Lua error in package.lua at line 80: module 'strict' not found.. Also in Lecture Notes in Mathematics, Combinatorics and Graph Theory, Springer-Verlag, New York, Vol. 885 (1980), p 500.
  3. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  4. ↑ *Lua error in package.lua at line 80: module 'strict' not found.
  5. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  6. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  7. ↑ 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

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