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.

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  A graph with frequency partition 6 = 3 + 2 + 1.
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  A bipartite graph with frequency partition 9 = 5 + 3 + 1.
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  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 identified; 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 ${\displaystyle (p-1)/2}$ 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:

• Line graphs
• Bipartite graphs ^[7]