A Möbius ladder, sometimes called a Möbius wheel (Jakobson and Rivin 1999), of order
is a simple graph obtained by introducing a twist
in a prism graph of order that is isomorphic to the circulant
graph.
Möbius ladders are sometimes denoted .
The 4-Möbius ladder is known as the Wagner graph. The -Möbius
ladder rung graph is isomorphic to the Haar graph.
Möbius ladders are Hamiltonian, graceful (Gallian 1987, Gallian 2018), and by construction, singlecross.
The Möbius ladders are also nontrivial biplanar
graphs. They are notunit-distance,
as can be proven by hand or using the "rhombus logic" of Globus and Parshall
(2020) and Alexeev et al. (2025) (pers. comm., B. Alexeev, Aug. 9,
2025).
The numbers of directed Hamiltonian cycles for , 4, ... are 12, 10, 16, 14, 20, 18,
24, ... (OEIS A124356), given by the closed
form
Alexeev, B.; Mixon, D. G.; and Parshall, H. "The Erdős Unit Distance Problem for Small Point Sets." 12 Feb 2025. https://arxiv.org/abs/2412.11914.Biggs,
N. L. Algebraic
Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, pp. 20-21,
1993.Gallian, J. "Labeling Prisms and Prism Related Graphs."
Congr. Numer.59, 89-100, 1987.Gallian, J. "Dynamic
Survey of Graph Labeling." Elec. J. Combin.DS6. Dec. 21,
2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Globus,
A. and Parshall, H. "Small Unit-Distance Graphs in the Plane." Bull.
Inst. Combin. Appl.90, 107-138, 2020.Godsil, C. and Royle,
G. Algebraic
Graph Theory. New York: Springer-Verlag, pp. 118 and 131, 2001.Hladnik,
M.; Marušič, D.; and Pisanski, T. "Cyclic Haar Graphs." Disc.
Math.244, 137-153, 2002.McSorley, J. P. "Counting
Structures in the Moebius Ladder." Disc. Math.184, 137-164, 1998.Jakobson,
D. and Rivin, I. "On Some Extremal Problems in Graph Theory." 8 Jul 1999.
http://arxiv.org/abs/math.CO/9907050.Read,
R. C. and Wilson, R. J. An
Atlas of Graphs. Oxford, England: Oxford University Press, pp. 263 and
270, 1998.Sloane, N. J. A. Sequence A124356
in "The On-Line Encyclopedia of Integer Sequences."
is a simple graph obtained by introducing a twist
in a prism graph of order 
that is isomorphic to the circulant
graph 
.
Möbius ladders are sometimes denoted 
.
-Möbius
ladder rung graph is isomorphic to the Haar graph 
.
, 4, ... are 12, 10, 16, 14, 20, 18,
24, ... (OEIS A124356), given by the closed
form
![|HC(n)|=2[(n+2)-(-1)^n].](/images/equations/MoebiusLadder/NumberedEquation1.svg)
-Möbius
ladder graph has independence polynomial
![I_n(x)=2^(-n)[-2^n(-x)^n+(x-sqrt(x(x+6)+1)+1)^n+(x+sqrt(x(x+6)+1)+1)^n].](/images/equations/MoebiusLadder/NumberedEquation2.svg)
-Möbius ladder is the prism
graph 
.
The graph square of 
is the circulant graph 
and its graph
cube is 
.
