A traceable graph is a graph that possesses a Hamiltonian path. Hamiltonian graphs are therefore traceable,
but the converse is not necessarily true. Graphs that are not traceable are said
to be untraceable.
The numbers of traceable graphs on , 2, ... are 1, 1, 2, 5, 18, 91, 734, ... (OEIS A057864),
where the singleton graph is conventionally considered traceable. The first few are
illustrated above, with a Hamiltonian path indicated
in orange for each.
-dimensional grid
graphs of arbitrary shape and dimension appear to be traceable, though a proof
of this assertion in its entirely does not seem to appear in the literature (cf.
Simmons 1978, Hedetniemi et al. 1980, Itai et al. 1982, Zamfirescu
and Zamfirescu 1992).
The following table lists some named graphs that are traceable but not Hamiltonian.
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, 2, ... are 1, 1, 2, 5, 18, 91, 734, ... (OEIS A057864),
where the singleton graph 
is conventionally considered traceable. The first few are
illustrated above, with a Hamiltonian path indicated
in orange for each.
-dimensional grid
graphs of arbitrary shape and dimension appear to be traceable, though a proof
of this assertion in its entirely does not seem to appear in the literature (cf.
Simmons 1978, Hedetniemi et al. 1980, Itai et al. 1982, Zamfirescu
and Zamfirescu 1992).
-Dimensional Rectangular Lattices Are Hamilton-Laceable."
In