Does an optimal path imply the heuristic is admissible?
No. 

Let us consider an extreme example. Let graph $G$ contain only two nodes, the starting node $s$ and the destination node $t$. The distance of edge $(s,t)$ is 1. 

We have a heuristic function $h$, $h(s)=2$ and $h(t)=0$. $h$ is not admissible since $h(s)&gt;1$.

However, using $A^*$ search algorithm, we will end up with the path $s, t$, the unique and, hence, optimal path.



Exercise. Construct a counterexample where there are more than one path.