The smallest degree sum that yields potentially K_(r,r)-graphic sequences

We consider a variation of a classical Turan-type extremal problem as follows: Determine the smallest even integer α(Kr,r,n) such that every n-term graphic sequence π = (d1,d2,…,dn) with term sum α(π) = d1 + d2 +… + dn ≥ α(Kr,r, n) is potentially Kr,r-graphic, where Kr,r is an r × r complete bipartite graph, i.e. π has a realization G containing KT.r as its subgraph. In this paper, the values α(Kr,r,n) for even r and n ≥ 4r2 - r- 6 and for odd r and n≥4r2+3r-8 are determined.