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For a decreasing probability sequence $p(n)$, if $G(n,p(n))$ contains a Hamilton cycle with high probability, then $G(n,Cp(n))$ is pancyclic with high probability for every fixed sufficiently large $C$; three independent rounds suffice. One first exposes a Hamiltonian round and uses the independent sprinkled edges to create cycles of every shorter length.

A connected graph on $s$ vertices containing at least two cycles has a spanning tree together with at least two further edges. By Cayley formula, the number of labelled spanning trees on a fixed $s$-set is $s^{s-2}$. Hence the first moment method and a union bound show that the probability of such a component of order $s$ is at mostwhere $p=(1-\epsilon )/n$ and $C$ is absolute. Requiring that the chosen set be a component would only add absent-edge conditions, so omitting them is a valid upper bound.

ThereforeThus every component in the stated range is either a tree or a unicyclic component with high probability.

Solved by gpt-5.6-sol high.

The sharp Hamilton cycle threshold for the Erdős-Rényi model says thatonly above the window $np=\log n+\log \log n+o(1)$. The hypothesis therefore places $p(n)$ above that window. The Hamiltonicity-to-pancyclicity sprinkling principle then says that three independent $G(n,p(n))$ rounds contain every cycle graph $C_{\ell }$, $3\le \ell \le n$, with high probability: one round supplies a Hamilton cycle, while the other two supply the chords and short-cycle edges used to obtain all intermediate lengths.

The union of the three rounds has individual edge probabilityBy the standard monotone coupling, it is a subgraph of $G(n,3p(n))$. Since being pancyclic is an increasing graph property, the required probability tends to one. If $3p(n)>1$, interpret the latter parameter as $\min \{3p(n),1\}$, in which case the conclusion is immediate.

Solved by gpt-5.6-sol high.

For every fixed $\epsilon >0$, every component of $G(n,(1-\epsilon )/n)$ has $O_{\epsilon }(\log n)$ vertices with high probability. A breadth-first exploration is dominated by a branching process of mean $1-\epsilon$; its total progeny has an exponentially decreasing tail.