A Rao-type characterization for a sequence to have a realization containing an arbitrary subgraph H

Let G be an arbitrary spanning subgraph of the complete graph K r+1 on r+1 vertices and K r+1 . E(G) be the graph obtained from K r+1 by deleting all edges of G. A non-increasing sequence . = (d 1, d 2, ..., d n ) of nonnegative integers is said to be potentially K r+1 . E(G)-graphic if there is a graph on n vertices that has . as its degree sequence and contains K r+1 .E(G) as a subgraph. In this paper, a characterization of . that is potentially K r+1 . E(G)-graphic is given, which is analogous to the Erd.os-Gallai characterization of graphic sequences using a system of inequalities. This is a solution to an open problem due to Lai and Hu. As a corollary, a characterization of . that is potentially K s,t -graphic can also be obtained, where K s,t is the complete bipartite graph with partite sets of size s and t. This is a solution to an open problem due to Li and Yin.