The Erd\H{o}s-Faudree Problems and the Isolate-Free Core

Abstract

In 1981, Erd\H{o}s and Faudree asked whether there exists an infinite family of graphs $G_N$ on $N$ vertices with $\Delta(G_N)<N-1$ and $\sri(G_N)=1$, and whether every family with $|V(G_N)|=N$ and $\Delta(G_N)<c$ for some fixed constant $c$ must satisfy $\sri(G_N)\to 0$. 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 $\bigl(\sr(tK_2,G)\bigr)_{t\ge 1}$ depends only on the isolate-free core $\core(G)$. 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 $|V(G_N)|=N$ and $\sri(G_N)\to\alpha$; in particular there are connected graphs with $\Delta(G_N)=N-2$ and $\sri(G_N)\to 1$. For Problem~2 we prove a strengthened positive statement: if $\Delta(G_N)<c$ for a fixed constant $c$ and the isolate-free core of $G_N$ has order tending to infinity, then $\sri(G_N)\to 0$. In particular every connected bounded-degree family satisfies $\sri(G_N)\to 0$. Thus the original Erd\H{o}s-Faudree questions are resolved in their literal form, and the mechanism behind their connected and disconnected behavior is identified precisely.