Imperfect Graph

An imperfect graph  is a graph that is not perfect. Therefore, graphs  with

(1)

where  is the clique number and  is the chromatic number are imperfect. A weaker form using a bound of  states that a graph with

(2)

where  is the independence number is imperfect.

By the perfect graph theorem, applying the above to the graph complement gives that a graph  with

(3)

where  is the clique covering number is also imperfect.

A graph is also imperfect iff either the graph or its complement has a chordless cycle of odd order.

Families of imperfect graphs include:

1. cycle graphs 

2. fullerenes (which by definition contain an odd 5-cycle)

3. king graphs  with 

4. helm graphs  for odd 

5. wheel graphs 

A list of imperfect graphs on small numbers of vertices is implemented in the Wolfram Language as GraphData["Imperfect"].

The numbers of simple imperfect graphs on , 2, ... vertices are 0, 0, 0, 0, 1, 8, 138, 3459, ... (OEIS A187236).

The numbers of connected imperfect graphs on , 2, ... vertices are 0, 0, 0, 0, 1, 7, 129, 3312,... (OEIS A187237).

REFERENCES

Godsil, C. and Royle, G. "Imperfect Graphs." §7.6 in Algebraic Graph Theory. New York: Springer-Verlag, pp. 142-145, 2001.

Sloane, N. J. A. Sequences A187236 and A187237 in "The On-Line Encyclopedia of Integer Sequences."West, D. B. "Imperfect Graphs." Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 334-340, 2000.