The Erdős-Faudree Problems and the Isolate-Free Core

In 1981, Erdős and Faudree asked whether there exists an infinite family of graphs $G_N$ on $N$ vertices with $\mathrm{\Delta }(G_N)<N-1$ and $\text{sri}(G_N)=1$, and whether every family with $\mathrm{\mid }V(G_N)\mathrm{\mid }=N$ and $\mathrm{\Delta }(G_N)<c$ for some fixed constant $c$ must satisfy $\text{sri}(G_N)\to 0$. We show first that the literal forms of the two questions are controlled entirely by isolated vertices: for every nonempty graph $G$, the whole sequence $(\text{sr}(t{K}_2,G){)}_{t\ge 1}$ depends only on the isolate-free core $\text{core}(G)$. Consequently, Problem 1 has a positive answer and Problem~2 has a negative answer in exactly their original form. We then turn to the genuine content behind the two problems. For Problem 1 we study connected graphs and prove a complete limit theorem: for every $\alpha \in [0,1]$ there exists a family of connected bipartite graphs $G_N$ with $\mathrm{\mid }V(G_N)\mathrm{\mid }=N$ and $\text{sri}(G_N)\to \alpha$; in particular there are connected graphs with $\mathrm{\Delta }(G_N)=N-2$ and $\text{sri}(G_N)\to 1$. For Problem~2 we prove a strengthened positive statement: if $\mathrm{\Delta }(G_N)<c$ for a fixed constant $c$ and the isolate-free core of $G_N$ has order tending to infinity, then $\text{sri}(G_N)\to 0$. In particular every connected bounded-degree family satisfies $\text{sri}(G_N)\to 0$. Thus the original Erdős-Faudree questions are resolved in their literal form, and the mechanism behind their connected and disconnected behavior is identified precisely.