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

Abstract

Abstract Let S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}) , where a 1 , … , a m {a}_{1},\ldots ,{a}_{m} and b 1 , … , b n {b}_{1},\ldots ,{b}_{n} are two nonincreasing sequences of nonnegative integers. The pair S = ( a 1 , … , a m ; b 1 , … , b n ) 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 graph G = ( X ∪ Y , E ) G=\left(X\cup Y,E) such that a 1 , … , a m {a}_{1},\ldots ,{a}_{m} and b 1 , … , b n {b}_{1},\ldots ,{b}_{n} are the degrees of the vertices in X X and Y Y , respectively. In this case, G G is referred to as a realization of S S . Given a bigraphic pair S S , and a complete bipartite graph K s , t {K}_{s,t} , we say that S S is a potentially K s , t {K}_{s,t} -bigraphic pair if some realization of S S contains K s , t {K}_{s,t} as a subgraph (with s s vertices in the part of size m m and t t in the part of size n n ). Ferrara et al. (Potentially H-bigraphic sequences, Discuss. Math. Graph Theory 29 (2009), 583–596) defined σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) to be the minimum integer k k such that every bigraphic pair S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};{b}_{1},\ldots ,{b}_{n}) with σ ( S ) = a 1 + ⋯ + a m ≥ k \sigma \left(S)={a}_{1}+\cdots +{a}_{m}\ge k is a potentially K s , 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. determined σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) for n ≥ m ≥ 9 s 4 t 4 n\ge m\ge 9{s}^{4}{t}^{4} . In this paper, we further determine σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) for n ≥ m ≥ s n\ge m\ge s and n + m ≥ 2 t 2 + t + s n+m\ge 2{t}^{2}+t+s . As two corollaries, if n ≥ m ≥ t 2 + t + s 2 n\ge m\ge {t}^{2}+\frac{t+s}{2} or if n ≥ m ≥ s n\ge m\ge s and n ≥ 2 t 2 + t n\ge 2{t}^{2}+t , the values σ ( K s , 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.