The above graph has the following properties :
- $1$) Every vertex is start vertex of some hamiltonian path.
- $2$) It contains no hamiltonian cycle.
- $3$) It has no cycle of length $3$.
- $4$) It is planar.
- $5$) It has at least $3$ vertices.
Property $5$ is only listed to avoid the $K_2$
I conjecture that this is the smallest graph with these properties.
The smallest graph fulfilling $1$) and $2$) and $5$) seems to have $9$ nodes, if $3$) also is required, then $10$ nodes seems to be the minimum. The example with $9$ nodes, which is introduced in the question Is there a name for graphs with the following property, is planar.
Is this the smallest graph with the properties $1-5$ ?