  In this paper, 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
the both 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
paper, 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 arose naturally in
statistical settings.
