Pseudo-h-hamiltonian graph

Pseudo-[math]\displaystyle{ h }[/math]-hamiltonian graph --- псевдо-[math]\displaystyle{ h }[/math]-гамильтонов граф.

For an integer [math]\displaystyle{ h \geq 1 }[/math], an undirected graph [math]\displaystyle{ G = (V,E) }[/math] is a pseudo-[math]\displaystyle{ h }[/math]-hamiltonian graph, if there exists a circular sequence of [math]\displaystyle{ h\cdot |V| }[/math] vertices such that the following properties hold:

(1) every vertex of [math]\displaystyle{ G }[/math] appears precisely [math]\displaystyle{ h }[/math] times in the sequence, and

(2) any two consecutive vertices in the sequence are adjacent in [math]\displaystyle{ G }[/math].

A sequence with these properties will be termed a pseudo-[math]\displaystyle{ h }[/math]-hamiltonian cycle. In this sense, pseudo-1-hamiltonian corresponds to the standard notion hamiltonian, and a pseudo-1-hamiltonian cycle is just a hamiltonian cycle. The pseudo-hamiltonicity number [math]\displaystyle{ ph(G) }[/math] of the graph [math]\displaystyle{ G }[/math] is the smallest integer [math]\displaystyle{ h \geq 1 }[/math] for which [math]\displaystyle{ G }[/math] is pseudo-[math]\displaystyle{ h }[/math]-hamiltonian; in case no such [math]\displaystyle{ h }[/math] exists, [math]\displaystyle{ ph(G) = \infty }[/math]. A graph [math]\displaystyle{ G }[/math] with finite [math]\displaystyle{ ph(G) }[/math] is called pseudo-hamiltonian. Pseudo-[math]\displaystyle{ h }[/math]-hamiltonicity is a non-trivial graph property. E.g., for every [math]\displaystyle{ h \geq 2 }[/math], the graph [math]\displaystyle{ G_{h} }[/math] that results from gluing together [math]\displaystyle{ h }[/math] triangles at one of their vertices is pseudo-[math]\displaystyle{ h }[/math]-hamiltonian but it is not pseudo-[math]\displaystyle{ (h-1) }[/math]-hamiltonian.