Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\le k\le n$. Estimate $h(n)$. In particular, is it true that$$h(n)\ge {\log }_{2}n+{\log }_{\ast }n-O(1),$$where ${\log }_{\ast }n$ is the iterated logarithmic function?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Your function $h(n)$ is exactly the “extra edges beyond a Hamilton cycle” needed to build a **pancyclic** $n$-vertex graph [[nomath]](a graph containing a cycle of every length $3,4,\dots,n$)[[/nomath]]. Equivalently, if $m(n)$ denotes the minimum number of edges in an $n$-vertex pancyclic graph, then [ m(n)=n+h(n). ] …

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.