| In this dissertation, we survey different notions of balance in a graph, with respect to the degress of the vertices in various subgraphs, and then, combining some of these concepts, we define a four-fold notion of balance with respect to a partition of the vertices of a graph. We define what it means for a graph G to be an amalgamation of a graph H, then what it means for G to be an outline graph (a graph which "looks like" the amalgamation of some simple graph, without that target graph being specified). We then ask, if given an outline G, when does there exist an actual simple graph H having G as an amalgamation, and exhibiting our four-fold balance with respect to the vertex partition given as part of the definition of G.; Next, we define what it means to be an outline edge-coloring of a graph, specifically of the complete graph on vertex set V, denoted KV. We ask the question, if given an outline edge-coloring of KV, call it G, then when does there exist an actual edge-colored copy of KV, call it H, such that G is an amalgamation of H and also such that each color class (i.e. spanning subgraph induced by the edges of a single color) exhibits our four-fold balance. The main result of this paper completely settles this question. |
