Path of length $n$ but no Hamilton cycle [closed]

Question

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.

Answer 1

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 may be of particular interest to you.

Answer 2

For $n\geq 5$: connect $i$ to $i+1$ for $1\leq i\leq n-1$, connect 1 to 3 and connect 3 to $n$. Every cycle that wants to visit each vertex, has to visit vertex 3 at least twice.