Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Results of Lovasz (1972) and Padberg (1974) imply that partitionable graphs contain all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture. A recursive method of generating partitionable graphs was suggested by Chvatal, Graham, Perold, and Whitesides (1979). Results of Sebo (1996) entail that Berge's conjecture holds for all the partitionable graphs obtained by this method. Here  suggest a more general recursion. Computer experiments show that it generates all the partitionable graphs with ω=3,α ≤ 9 (and  conjecture that the same will hold for bigger α, too) and many but not all for (ω,α)=(4,4) and (4,5). Here, α and ω are respectively the clique and stability numbers of a partitionable graph, that is the numbers of vertices in its maximum cliques and stable sets. All the partitionable graphs generated by our method contain a critical ω-clique, that is an ω-clique which intersects only 2ω-2 other ω-cliques. This property might imply that in our class there are no counterexamples to Berge's conjecture (cf. Sebo (1996)), hover this question is still open. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 259–285, 2002
['Endre Boros', 'Vladimir Gurvich', 'Stefan Hougardy']
I am writing a research paper on a idea Recursive generation of partitionable graphs
[{"from": "human", "value": "I am writing a research paper on a idea Recursive generation of partitionable graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Results of Lovasz (1972) and Padberg (1974) imply that partitionable graphs contain all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture. A recursive method of generating partitionable graphs was suggested by Chvatal, Graham, Perold, and Whitesides (1979). Results of Sebo (1996) entail that Berge's conjecture holds for all the partitionable graphs obtained by this method. Here  suggest a more general recursion. Computer experiments show that it generates all the partitionable graphs with \u03c9=3,\u03b1 \u2264 9 (and  conjecture that the same will hold for bigger \u03b1, too) and many but not all for (\u03c9,\u03b1)=(4,4) and (4,5). Here, \u03b1 and \u03c9 are respectively the clique and stability numbers of a partitionable graph, that is the numbers of vertices in its maximum cliques and stable sets. All the partitionable graphs generated by our method contain a critical \u03c9-clique, that is an \u03c9-clique which intersects only 2\u03c9-2 other \u03c9-cliques. This property might imply that in our class there are no counterexamples to Berge's conjecture (cf. Sebo (1996)), hover this question is still open. \u00a9 2002 Wiley Periodicals, Inc. J Graph Theory 41: 259\u2013285, 2002"}]