On multigraphic and potentially multigraphic sequences

An r-graph(or a multigraph) is a loopless graph in which no two vertices are joined by more than r edges. An r-complete graph on n vertices, denoted by K_n^(r) , is an r-graph on n vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π = (d_1, d_2, ..., d_n) of non-negative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially S_L;M^(r) -graphic if it has a realization containing S_L;M^(r) as a subgraph. We obtain conditions for an r-graphic sequence to be potentially S(r) L;M-graphic. These are generalizations from split graphs to p-tuple r-split graph.