A variation of a classical Turلn-type extremal problem

A variation of a classical Turán-type extremal problem (Erdős on Graphs: His Legacy of Unsolved Problems (1998) p. 36) is considered as follows: determine the smallest even integer σ(Kr,s,n) such that every n-term graphic non-increasing sequence π=(d1,d2,…,dn) with term sum σ(π)=d1+d2+⋯+dn≥σ(Kr,s,n) has a realization G containing Kr,s as a subgraph, where Kr,s is a r×s complete bipartite graph. In this paper, we determine σ(Kr,s,n) exactly for every fixed s≥r≥3 when n≥n0(r,s), where m=[(r+s+1)24] andn0(r,s)=m+3s2−2s−6,if s≤2r and s is even,m+3s2+2s−8,if s≤2r and s is odd,m+2s2+(2r−6)s+4r−8,if s≥2r+1.