On the potential function [formula omitted] of an arbitrary bipartite graph [formula omitted]

Abstract

Let π=(f1,…,fm;g1,…,gn), where f1,…,fm and g1,…,gn are two nonincreasing sequences of nonnegative integers. The pair π=(f1,…,fm;g1,…,gn) is said to be a bigraphic pair if there is a simple bipartite graph G[X,Y] with vertex bipartition (X,Y) such that f1,…,fm and g1,…,gn are the degrees of the vertices in X and Y, respectively. In this case, G is referred to as a realization of π. For a given bipartite graph H=H[X,Y] with |X|=s and |Y|=t, we say that π is a potentiallyH-bigraphic pair if π has a realization G containing H as a subgraph with the s vertices of X in the part of G of size m and the t vertices of Y in the part of G of size n. Let σ(H,m,n) denote the minimum integer k such that every bigraphic pair π=(f1,…,fm;g1,…,gn) with σ(π)≥k is a potentially H-bigraphic pair, where σ(π)=f1+⋯+fm. The parameter σ(H,m,n) is known as the potential function of H, and can be viewed as a degree sequence variant of the classical extremal function ex(H,m,n) as introduced by Erdős et al. Ferrara et al. determined σ(Ks,t,m,n) for n≥m≥9s4t4, σ(Pℓ,m,n) for n≥m≥⌈ℓ2⌉ and σ(C2t,m,n) for n≥m≥2(t+1). In this paper, for an arbitrary bipartite graph H, we firstly give a construction that yields a lower bound on σ(H,m,n). Then, we determine σ(As,td,m,n) for n≥2s2t2, where As,td is the graph obtained from Ks−1,t by adding a new vertex that is adjacent to d vertices of the part of size t. Finally, we investigate the precise behavior of σ(H,m,n) for an arbitrary bipartite graph H, and determine σ(H,m,n) for n≥max{m(t−d)+(s−1)(d−1),2s2t2}, where d=min{dH(x)|x∈X}.