Möbius ladder

In graph theory, the Möbius ladder Mn is a cubic circulant graph with an even number n of vertices, formed from an n-cycle by adding edges (called 'rungs') connecting opposite pairs of vertices in the cycle. It is so-named because (with the exception of M6 = K3,3) Mn has exactly n/2 4-cycles which link together by their shared edges to form a topological Möbius strip. Möbius ladders were named and first studied by Guy and Harary (1967). In graph theory, the Möbius ladder Mn is a cubic circulant graph with an even number n of vertices, formed from an n-cycle by adding edges (called 'rungs') connecting opposite pairs of vertices in the cycle. It is so-named because (with the exception of M6 = K3,3) Mn has exactly n/2 4-cycles which link together by their shared edges to form a topological Möbius strip. Möbius ladders were named and first studied by Guy and Harary (1967). Every Möbius ladder is a nonplanar apex graph, meaning that it cannot be drawn without crossings in the plane but removing one vertex allows the remaining graph to be drawn without crossings. Möbius ladders have crossing number one, and can be embedded without crossings on a torus or projective plane. Thus, they are examples of toroidal graphs. Li (2005) explores embeddings of these graphs onto higher genus surfaces. Möbius ladders are vertex-transitive – they have symmetries taking any vertex to any other vertex – but (again with the exception of M6) they are not edge-transitive. The edges from the cycle from which the ladder is formed can be distinguished from the rungs of the ladder, because each cycle edge belongs to a single 4-cycle, while each rung belongs to two such cycles. Therefore, there is no symmetry taking a cycle edge to a rung edge or vice versa. When n ≡ 2 (mod 4), Mn is bipartite. When n ≡ 0 (mod 4), it is not bipartite. The endpoints of each rung are an even distance apart in the initial cycle, so adding each rung creates an odd cycle.In this case, because the graph is 3-regular but not bipartite, by Brooks' theorem it has chromatic number 3. De Mier & Noy (2004) show that the Möbius ladders are uniquely determined by their Tutte polynomials. The Möbius ladder M8 has 392 spanning trees; it and M6 have the most spanning trees among all cubic graphs with the same number of vertices. However, the 10-vertex cubic graph with the most spanning trees is the Petersen graph, which is not a Möbius ladder. The Tutte polynomials of the Möbius ladders may be computed by a simple recurrence relation. Möbius ladders play an important role in the theory of graph minors. The earliest result of this type is a theorem of Klaus Wagner (1937) that graphs with no K5 minor can be formed by using clique-sum operations to combine planar graphs and the Möbius ladder M8; for this reason M8 is called the Wagner graph. Gubser (1996) defines an almost-planar graph to be a nonplanar graph for which every nontrivial minor is planar; he shows that 3-connected almost-planar graphs are Möbius ladders or members of a small number of other families, and that other almost-planar graphs can be formed from these by a sequence of simple operations. Maharry (2000) shows that almost all graphs that do not have a cube minor can be derived by a sequence of simple operations from Möbius ladders.