Stability of the Potential Function

A graphic sequence $\pi$ is potentially $H$-graphic if there is some realization of $\pi$ that contains $H$ as a subgraph. The Erdos--Jacobson--Lehel problem asks one to determine $\sigma(H,n)$, the minimum even integer such that any $n$-term graphic sequence $\pi$ with sum at least $\sigma(H,n)$ is potentially $H$-graphic. The parameter $\sigma(H,n)$ is known as the potential function of $H$, and can be viewed as a degree sequence variant of the classical extremal function ${ex}(n,H)$. Recently, Ferrara et al. [Combinatorica 36 (2016), pp. 687--702] determined $\sigma(H,n)$ asymptotically for all $H$, which is analogous to the Erdos--Stone--Simonovits theorem that determines ${ex}(n,H)$ asymptotically for nonbipartite $H$. In this paper, we investigate a stability concept for the potential number, inspired by Simonovits' classical result on the stability of the extremal function. We first define a notion of stability for the potential number that is a natural analogue to the stability given by Simonovits...