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Is it true that for perfect graphs the following holds:

A maximum size of an independent set in a graph $G$ equals to the chromatic number of $\overline{G}$, where $\overline{G}$ is the complement of $G$ of course.

If yes, what would be the reference?

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By the perfect graph theorem (conjectured by Claude Berge in 1961 and proved by László Lovász in 1972 [L]), a graph is perfect iff its complement is perfect. Therefore a chromatic number of $\overline{G}$ equals to its clique number, which equals to an independence number of $G$.

References

[L] L. Lovász, A characterization of perfect graphs, J. combin. theory ser. B 13 (1972) 95–98.

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