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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 complementhas 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..

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