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Amalgamations of connected $$k$$-factorizations. (English) Zbl 1033.05084
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.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C15 Coloring of graphs and hypergraphs
##### Keywords:
edge-coloring; amalgamation; graph factorization; complete graph
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