Amalgamations of connected \(k\)-factorizations. (Q1400963)

Let \(G\) be an edge-colored graph on the set of colors \(\{ 1,2,\ldots,n\}\), and let \(P\) be a partition of \(V(G)\). The \(P\)-amalgamation of \(G\) is the \(n\)-edge-colored graph with vertex set \(P\), in which \(p\in P\) is incident with \(y\) loops colored \(i\) if and only if there are \(y\) loops or edges in \(G\) that join vertices in \(p\), and in which the number of edges colored \(i\) joining \(p_1,p_2\in P\), \(p_1\not= p_2\), is the number of edges colored \(i\) in \(G\) that join vertices in \(p_1\) to vertices in \(p_2\). We say that \(H\) is an amalgamation of \(G\) if it is the \(P\)-amalgamation of \(G\) for some partition \(P\) of \(V(G)\). This paper contains two main results: (1) a necessary and sufficient condition for a graph with exactly one amalgamated vertex to be the amalgamation of a \(k\)-factorization of the complete graph \(K_{kn+1}\) in which each \(k\)-factor is connected, (2) a necessary and sufficient condition for a given edge-colored complete graph \(K_t\) to be embedded in a \(k\)-factorization of \(K_{kn+1}\) in which each \(k\)-factor is connected.