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Multiply balanced edge colorings of multigraphs. (English) Zbl 1244.05085
Summary: In this article, a theorem is proved that generalizes several existing amalgamation results in various ways. The main aim is to disentangle a given edge-colored amalgamated graph so that the result is a graph in which the edges are shared out among the vertices in ways that are fair with respect to several notions of balance (such as between pairs of vertices, degrees of vertices in both the graph and in each color class, etc.). The connectivity of color classes is also addressed. Most results in the literature on amalgamations focus on the disentangling of amalgamated complete graphs and complete multipartite graphs. Many such results follow as immediate corollaries to the main result in this article, which addresses amalgamations of graphs in general, allowing for example the final graph to have multiple edges. A new corollary of the main theorem is the settling of the existence of Hamilton decompositions of the family of graphs $$K(a_{1}, \dots , a_{p};\lambda _{1}, \lambda _{2})$$; such graphs arise naturally in statistical settings.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C45 Eulerian and Hamiltonian graphs
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