There are several definitions of "almost Hamiltonian" 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.
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's 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 a -cycle and therefore not almost Hamiltonian are summarized in the following table.