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• In this paper, we present a pair of lower and upper bounds of R g-conditional diagnosability of general networks under the PMC mode. • An improved lower bound of R g-conditional diagnosability of hypercubes and a pair of lower and upper bounds of R g-conditional diagnosability of exchanged hypercubes are obtained. The R g-conditional diagnosability is a new generalization of conditional diagnosability, which restricts that every vertex contains at least g fault-free neighbors. Recently, Wang et al. investigated the R g-conditional diagnosability of the n -dimensional hypercube under the PMC model, and showed a lower bound of its R g-conditional diagnosability. In this paper, we investigate the R g-conditional diagnosability of general networks under the PMC model, and present the lower and upper bounds of R g-conditional diagnosability of networks under some reasonable conditions. Applying our results, an improved lower bound of R g-conditional diagnosability of hypercubes and the lower and upper bounds of the R g-conditional diagnosability of exchanged hypercubes are given.