  Multi-layer networks or multiplex networks are generally considered as the
networks that have the same set of vertices but different types of edges.
Multi-layer networks are especially useful when describing the systems with
several kinds of interactions. In this paper we study the analytical solution
of $\textbf{k}$-core pruning process on multi-layer networks. $k$-core
decomposition is a widely used method to find the dense core of the network.
Previously the Nonbacktracking Expand Branch (NBEB) is found to be able to
easily derive the exact analytical results in the $k$-core pruning process.
Here we further extend this method to solve the $\textbf{k}$-core pruning
process on multi-layer networks by designing a variation of the method called
Multicolor Nonbacktracking Expand Branch (MNEB). Our results show that, given
any initial multi-layer network, Multicolor Nonbacktracking Expand Branch can
offer the exact solution for each intermediate state of the pruning process,
these results do not only apply to uncorrelated network, but also apply to
networks with either interlayer correlations or in-layer correlations.
