In this paper, we introduce the enough g-pairs property for principal coefficients cluster algebras, which can be understood as a strong version of the sign-coherence of the G-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enough g-pairs property. As applications, we prove some long standing conjectures in cluster algebras, including a conjecture on denominator vectors and a conjecture on exchange graphs (see Conjectures 1, 2 below). In addition, we give a criterion to distinguish whether particular cluster variables belong to one common cluster for any skew-symmetrizable cluster algebra. As a corollary, we prove a conclusion which was conjectured by Fomin et al., cf.

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