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LetS=(a1,…,am;b1,…,bn)S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}), wherea1,…,am{a}_{1},\ldots ,{a}_{m}andb1,…,bn{b}_{1},\ldots ,{b}_{n}are two nonincreasing sequences of nonnegative integers. The pairS=(a1,…,am;b1,…,bn)S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n})is said to be a  bigraphic pair  if there is a simple bipartite graphG=(X∪Y,E)G=\left(X\cup Y,E)such thata1,…,am{a}_{1},\ldots ,{a}_{m}andb1,…,bn{b}_{1},\ldots ,{b}_{n}are the degrees of the vertices inXXandYY, respectively. In this case,GGis referred to as a  realization  ofSS. Given a bigraphic pairSS, and a complete bipartite graphKs,t{K}_{s,t}, we say thatSSis a  potentiallyKs,t{K}_{s,t}- bigraphic pair  if some realization ofSScontainsKs,t{K}_{s,t}as a subgraph (withssvertices in the part of sizemmandttin the part of sizenn). Ferrara et al. ( Potentially H-bigraphic sequences , Discuss. Math. Graph Theory  29  (2009), 583–596) definedσ(Ks,t,m,n)\sigma \left({K}_{s,t},m,n)to be the minimum integerkksuch that every bigraphic pairS=(a1,…,am;b1,…,bn)S=\left({a}_{1},\ldots ,{a}_{m};{b}_{1},\ldots ,{b}_{n})withσ(S)=a1+⋯+am≥k\sigma \left(S)={a}_{1}+\cdots +{a}_{m}\ge kis a potentiallyKs,t{K}_{s,t}-bigraphic pair. This problem can be viewed as a “potential” degree sequence relaxation of the (forcible) Turán problem. Ferrara et al. determinedσ(Ks,t,m,n)\sigma \left({K}_{s,t},m,n)forn≥m≥9s4t4n\ge m\ge 9{s}^{4}{t}^{4}. In this paper, we further determineσ(Ks,t,m,n)\sigma \left({K}_{s,t},m,n)forn≥m≥sn\ge m\ge sandn+m≥2t2+t+sn+m\ge 2{t}^{2}+t+s. As two corollaries, ifn≥m≥t2+t+s2n\ge m\ge {t}^{2}+\frac{t+s}{2}or ifn≥m≥sn\ge m\ge sandn≥2t2+tn\ge 2{t}^{2}+t, the valuesσ(Ks,t,m,n)\sigma \left({K}_{s,t},m,n)are determined completely. These results give a solution to a problem due to Ferrara et al. and a solution to a problem due to Yin and Wang.

Solutions to problems about potentially K s , t -bigraphic pair