Amalgamations of connected k-factorizations

In this paper, necessary and sufficient conditions are found for a graph with exactly one amalgamated vertex to be the amalgamation of a k-factorization of Kkn+1 in which each k-factor is connected. From this result, necessary and sufficient conditions for a given edge-coloured Kt to be embedded in a connected k-factorization of Kkn+1 are deduced.