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# Bold Conjecture

A pair of vertices of a graph is called an -critical pair if , where denotes the graph obtained by adding the edge to and is the clique number of . The -critical pairs are never edges in . A maximal stable set of is called a forced color class of if meets every -clique of , and -critical pairs within form a connected graph.

In 1993, G. Bacsó conjectured that if is a uniquely -colorable perfect graph, then has at least one forced color class. This conjecture is called the bold conjecture, and implies the strong perfect graph theorem. However, a counterexample of the conjecture was subsequently found by Sakuma (1997).

Clique Number, Strong Perfect Graph Theorem

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## References

Sakuma, T. "A Counterexample to the Bold Conjecture." J. Graph Th. 25, 165-168, 1997.Sebő, A. "On Critical Edges in Minimal Perfect Graphs." J. Combin. Th. B 67, 62-85, 1996.

Bold Conjecture

## Cite this as:

Weisstein, Eric W. "Bold Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BoldConjecture.html