Existence of Decompositions of Multipartite Graphs, some Gregarious and some Fair

Abstract: 

A G-decomposition of H is a partition P(E) of the edges of H into sets, each of which induces a copy of the graph G. A G-decomposition of H is said to be gregarious with respect to a partition P(V) of the vertices of H if each copy of G in the decomposition contains vertices from as many different parts of P(V) as possible. A G-decomposition of H is said to be fair with respect to P(V) if the edges between vertices in each pair of parts are shared as evenly as possible among the elements of P(E). In this talk, the existence of decompositions of H, some being gregarious and some being fair, is discussed in the case where two vertices of H are joined by x edges if they occur in the same part of P(V), and by y edges if they occur in different parts of P(V).