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 sufficient condition using the lower
bound 
is
 |
(2)
|
where 
is the graph order and 
is the independence
number.
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.
By the strong perfect graph theorem, a graph is imperfect iff it or its graph complement contains a chordless cycle of odd graph order at least 5 (Chudnovsky et al. 2006).
Families of imperfect graphs include:
1. cycle graphs 
for 
2. fullerenes (which by definition contain an odd 5-cycle)
3. king graphs 
with 
4. helm graphs 
for odd 
5. wheel graphs 
for 
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).
