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.
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.
[1] Frequency partitions of various graph families are completely identified; frequency partitions of many families of graphs are not identified.
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