The Erdős-Faudree Problems and the Isolate-Free Core
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908973943226368 |
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| author | Mao, Yaping |
| author_facet | Mao, Yaping |
| contents | In 1981, Erdős and Faudree asked whether there exists an infinite family of graphs $G_N$ on $N$ vertices with $Δ(G_N)<N-1$ and $\sri(G_N)=1$, and whether every family with $|V(G_N)|=N$ and $Δ(G_N)<c$ 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 $α\in[0,1]$ there exists a family of connected bipartite graphs $G_N$ with $|V(G_N)|=N$ and $\sri(G_N)\toα$; in particular there are connected graphs with $Δ(G_N)=N-2$ and $\sri(G_N)\to 1$. For Problem~2 we prove a strengthened positive statement: if $Δ(G_N)<c$ 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 Erdős-Faudree questions are resolved in their literal form, and the mechanism behind their connected and disconnected behavior is identified precisely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16012 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Erdős-Faudree Problems and the Isolate-Free Core Mao, Yaping Combinatorics In 1981, Erdős and Faudree asked whether there exists an infinite family of graphs $G_N$ on $N$ vertices with $Δ(G_N)<N-1$ and $\sri(G_N)=1$, and whether every family with $|V(G_N)|=N$ and $Δ(G_N)<c$ 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 $α\in[0,1]$ there exists a family of connected bipartite graphs $G_N$ with $|V(G_N)|=N$ and $\sri(G_N)\toα$; in particular there are connected graphs with $Δ(G_N)=N-2$ and $\sri(G_N)\to 1$. For Problem~2 we prove a strengthened positive statement: if $Δ(G_N)<c$ 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 Erdős-Faudree questions are resolved in their literal form, and the mechanism behind their connected and disconnected behavior is identified precisely. |
| title | The Erdős-Faudree Problems and the Isolate-Free Core |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.16012 |