Möbius ladder

In graph theory, the Möbius ladder Mn, for even numbers n, is formed from an n-cycle by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the exception of M6 (the utility graph K3,3), Mn has exactly n/2 four-cycles which link together by their shared edges to form a topological Möbius strip. Möbius ladders were named and first studied by . For every even n > 4, the Möbius ladder Mn 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. These graphs have crossing number one, and can be embedded without crossings on a torus or projective plane. Thus, they are examples of toroidal graphs. 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 (with the exceptions of M4 and 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. 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.