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The colored branch theorem (Minty 1960 ) is a result in graph theory, which essentially says that the existence (resp., nonexistence) of a certain loop immediately implies the nonexistence (resp., existence) of a certain cutset. Its relevance and use in circuit theory, however, has only recently been recognized. Since it is expected that many more applications in circuit theory will follow, the theorem is interpreted and proved in a network setting. Many graph-theoretic corollaries are derived, which may facilitate later use. It is illustrated that many results in circuit theory can he simplified or given a simpler proof using this theorem and its corollaries.