A graph is Hamilton-connected if, for any vertices
, there is a Hamiltonian path from
. In most cases the Harary graphs are circulants, or have a circulant as an edge subgraph (see  or  for the explicit construction). For example, if
is even or
is even, then
is a circulant graph. And if
is odd, then
has a circulant as an edge subgraph. When
, the circulants that arise are connected, not bipartite, and not a cycle, and therefore the Chen–Quimpo theorem  on Hamilton-connectedness of such circulants applies: such graphs are Hamilton-connected.
is different, because the circulant graphs that underlie the Harary graphs are not all Hamilton-connected.
is a bipartite circulant and so is not Hamilton-connected, though it is Hamilton-laceable, by the Chen–Quimpo theorem.
can be proved to be not Hamilton-connected: there is no path from the unique vertex of maximum degree to the vertex two steps away in the natural circular order.
is the reason for this Demonstration, which shows the paths that arise in a proof that these graphs are Hamilton-connected. There are several different cases based on the parity of the starting vertex and the position of the target vertex.
 F. Harary, "The Maximum Connectivity of a Graph," Proceedings of the National Academy of Science
(7), 1962 pp. 1142–1146. www.jstor.org/stable/71730
 D. B. West, Introduction to Graph Theory
, 2nd ed., Upper Saddle River, NJ: Prentice-Hall, 2001 pp. 150–151.