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A graph G is called Hamiltonian if there is a Hamiltonian cycle in G. The conditional edge-fault Hamiltonicity of a Hamiltonian graph G is the largest k such that after removing k faulty edges from G, provided that each node is incident to at least two fault-free edges, the resulting graph contains a Hamiltonian cycle. In this paper, we sketch common properties of a class of networks, called matching composition networks (MCNs), such that the conditional edge-fault hamiltonicity of MCNs can be determined from the found properties. We then apply our technical theorems to determine conditional edge-fault hamiltonicities of several multiprocessor systems, including n-dimensional crossed cubes, n-dimensional twisted cubes, n-dimensional locally twisted cubes, n-dimensional generalized twisted cubes, and n-dimensional hyper Petersen networks. Moreover, we also demonstrate that our technical theorems can be applied to network construction.