Let G be a simple undirected Graph with the following property : For every pair of vertices $(u,v)$, there is a hamilton path from $u$ to $v$. It is clear that all complete graphs have this property and that the graph must have a hamilton circle to have this property.
A hamilton path is a path visiting EVERY vertex exactly once. A hamilton circle is a circle containing EVERY vertex.
Is there a name for such graphs ?
Which graphs beside the complete ones have this property (Is there a nice criterion) ?
What is the least possible number of edges for a graph with $n$ vertices to have this property ?