The term "almost Hamiltonian graph" has several definitions in use.
As defined by Punnim et al. (2007), an almost Hamiltonian graph is a graph on 
nodes having Hamiltonian number 
.
As defined by Sanders (1987), a graph 
with 
vertices is almost Hamiltonian if every subset of 
vertices is contained in a circuit, a definition which is
similar to that for a maximally nonhamiltonian
graph.
This work adopts the definition of Punnim et al. (2007).
The numbers of almost Hamiltonian graphs of this type on 
, 2, ... vertices are 0, 0, 1, 1, 7, 28, 257, 2933,
... (OEIS A185360), the first few of which
are illustrated above.
Under this definition, a connected graph 
on 
vertices is almost Hamiltonian iff 
is nonhamiltonian and has a Hamiltonian
path whose endpoints have a common neighbor.
Under the same definition, if an almost Hamiltonian graph 
is bipartite with nonempty
bipartition classes of sizes 
and 
,
then its Hamiltonian number 
is even and satisfies 
, where 
is the vertex count of
. Equivalently, 
.
Since a tree has Hamiltonian number 
, an almost Hamiltonian tree
satisfies 
, giving 
.
Since the 3-path graph 
is the only tree on three nodes,
it is also the only almost Hamiltonian tree.
Hypohamiltonian graphs are almost Hamiltonian. Many (or perhaps even all?) snarks are almost Hamiltonian.
Punnim et al. (2007) showed that, for 
,
the generalized Petersen graph 
is almost Hamiltonian iff 
.
Punnim et al. (2007) proved that there exists an almost Hamiltonian cubic graph for every even order 
.
The Petersen graph is the unique almost Hamiltonian
cubic graph on 10 vertices, and Tietze graph is the
unique almost Hamiltonian cubic graph on 12 vertices (Punnim et al. 2007).
Punnim et al. (2007) proved that a cubic nonhamiltonian graph on 
vertices is almost Hamiltonian iff
it is 2-connected and contains a cycle of length 
. Examples of 2-connected cubic nonhamiltonian graphs
on 
vertices lacking an 
-cycle and therefore not almost Hamiltonian are summarized
in the following table.
 | graph |
| 14 | 14-cubic graph 120 |
| 30 | triangle replacement of Petersen graph  |
| 36 | 36-Zamfirescu graph |
| 50 | Georges-Kelmans graph |
| 56 | 56-Ellingham-Horton graph |
