What is an example of a simple graph $G = (\{1,\ldots,n\}, E)$, where $n\in\mathbb{N}$ is a positive integer, with the following properties?


1. There is a path in $G$ of length $n$,
2. every vertex has at least $2$ neighbors, and
3. $G$ does *not* have a [Hamiltonian cycle](https://en.wikipedia.org/wiki/Hamiltonian_path).



Path of length $n$ but no Hamilton cycle
Assuming you mean a graph with a Hamiltonian path but no Hamiltonian cycle,


* The Petersen graph is a standard example.
* Any pendant-free graph with a Hamiltonian path and a bridge is an easy example; e.g. take two pendant-free graphs $G\_1$ and $G\_2$ with Hamiltonian paths (and optionally with Hamiltonian cycles). Let $s\_1, t\_1$ be endpoints of a Hamiltonian path in $G\_1$ and similarly $s\_2, t\_2$ in $G\_2$. Join them either by merging $s\_1$ and $s\_2$ or by adding a single edge $s\_1 - s\_2$.
* The class of [maximally non-Hamiltonian graphs](https://mathworld.wolfram.com/MaximallyNonhamiltonianGraph.html) may be of particular interest to you.


