On the potential function σ(H,m,n) of an arbitrary bipartite graph H

Abstract

Let $\pi =(f_1,\dots ,f_m;g_1,\dots ,g_n)$, where $f_1,\dots ,f_m$ and $g_1,\dots ,g_n$ are two nonincreasing sequences of nonnegative integers. The pair $\pi =(f_1,\dots ,f_m;g_1,\dots ,g_n)$ is said to be a bigraphic pair if there is a simple bipartite graph $G[X,Y]$ with vertex bipartition $(X,Y)$ such that $f_1,\dots ,f_m$ and $g_1,\dots ,g_n$ are the degrees of the vertices in $X$ and $Y$, respectively. In this case, $G$ is referred to as a realization of $\pi$. For a given bipartite graph $H=H[X,Y]$ with $\left|X\right|=s$ and $\left|Y\right|=t$, we say that $\pi$ is a potentially $H$-bigraphic pair if $\pi$ 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 $\sigma (H,m,n)$ denote the minimum integer $k$ such that every bigraphic pair $\pi =(f_1,\dots ,f_m;g_1,\dots ,g_n)$ with $\sigma \left(\pi \right)\ge k$ is a potentially $H$-bigraphic pair, where $\sigma \left(\pi \right)=f_1+\cdots +f_m$. The parameter $\sigma (H,m,n)$ 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 $\sigma (K_{s,t},m,n)$ for $n\ge m\ge 9s^4{t}^4$, $\sigma (P_{\ell },m,n)$ for $n\ge m\ge \lceil \frac{\ell }{2}\rceil$ and $\sigma (C_{2t},m,n)$ for $n\ge m\ge 2(t+1)$. In this paper, for an arbitrary bipartite graph $H$, we firstly give a construction that yields a lower bound on $\sigma (H,m,n)$. Then, we determine $\sigma (A_{s,t}^d,m,n)$ for $n\ge 2s^2{t}^2$, where $A_{s,t}^d$ is the graph obtained from $K_{s-1,t}$ by adding a new vertex that is adjacent to $d$ vertices of the part of size $t$. Finally, we investigate the precise behavior of $\sigma (H,m,n)$ for an arbitrary bipartite graph $H$, and determine $\sigma (H,m,n)$ for $n\ge max\{m(t-d)+(s-1)(d-1),2s^2{t}^2\}$, where $d=min\left\{d_H\left(x\right)\right|x\in X\}$.