Extremal Graph

In general, an extremal graph is the largest graph of order  which does not contain a given graph  as a subgraph (Skiena 1990, p. 143). Turán studied extremal graphs that do not contain a complete graph  as a subgraph.

One much-studied type of extremal graph is a two-coloring of a complete graph  of  nodes which contains exactly the number  of monochromatic forced triangles and no more (i.e., a minimum of  where  and  are the numbers of red and blue triangles). Goodman (1959) showed that for an extremal graph of this type,

(1)

This is sometimes known as Goodman's formula. Schwenk (1972) rewrote it in the form

(2)

sometimes known as Schwenk's formula, where  is the floor function. The first few values of  for , 2, ... are 0, 0, 0, 0, 0, 2, 4, 8, 12, 20, 28, 40, 52, 70, 88, ... (OEIS A014557).

REFERENCES

Goodman, A. W. "On Sets of Acquaintances and Strangers at Any Party." Amer. Math. Monthly 66, 778-783, 1959.

Schwenk, A. J. "Acquaintance Party Problem." Amer. Math. Monthly 79, 1113-1117, 1972.

Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 143, 1990.

Sloane, N. J. A. Sequence A014557 in "The On-Line Encyclopedia of Integer Sequences."