check of Möbius — Wikipedia

In graph theory, a branch of mathematics, the Möbius scale

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is a cubic graph formed from the graph cycle to

${displaystyle 2n}$

Summits by adding edges between the opposite summits of the cycle.

The graphs of this family are so named because, if we except

${displaystyle ML_{3}}$

^{[ 2 ]} ,

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has exactly

${displaystyle n}$

4 summits cycles ^{[ 3 ]} which, put together by their shared summits, form the equivalent of a ribbon of Möbius. The scales of Möbius were appointed and studied for the first time by Richard Guy and Frank Hairry in 1967 ^{[ 4 ]} .

Möbius scales are circulating graphs.

They are not planar graphs, but can be made planar by removing a single summit, which makes them apex graphs (in) . It is possible to draw it on a plane with a single cross and this number is minimal. It can be plunged (in) In a torus or a projective plan without crosses, it is therefore an example of Toroidal Graph. De-Ming Li explored diving of these graphs on superior order surfaces ^{[ 5 ]} .

The scales of Möbius are summary but are not transtitious edges (except

${displaystyle ML_{3}}$

): Each amount of the scale belongs to a single cycle of 4 peaks, while the bars of the scale each belong to two of these cycles.

The Brooks theorem and the fact that the graph is cubic guarantee that 3 colors are enough to color it. In fact, when

${displaystyle n}$

is peer, we need the three colors, and otherwise two colors are enough. In addition, Möbius scales are uniquely determined by their chromatic polynomials ^{[ 6 ]} .

Möbius scale

${displaystyle ML_{4}}$

has 392 covering trees. Her and

${displaystyle ML_{3}}$

have the most covering trees among all cubic graphs with the same number of summits ^{[ 7 ] , [ 8 ]} . However, it is not general. Indeed, the cubic graph at 10 summit having the most covering trees is the graph of Petersen, which is not a scale of Möbius.

Tuttes polynomials of Möbius scales can be calculated using a simple recurrence relationship ^{[ 9 ]} .

The scales of Möbius play an important role in the history of graph minors. The oldest result in this area is Klaus Wagner’s 1937 theorem saying that graphs without a minor can be formed using the sum operations (in) To combine planar graphs and Möbius scale

${displaystyle ML_{4}}$

. For this reason,

${displaystyle ML_{4}}$

Wagner graph is called ^{[ ten ]} .

Gubser (1996) defines a almost planar graph Like a non -planar graph in which any non -trivial minor is planar. He shows that almost planar graphs 3-connections are Möbius scales or members of a small number of other families and that other almost-plain graphs can be formed from a series of simple operations ^{[ 11 ]} .

John Maharry has shown that almost all graphs that do not have a cubic minor can be deducted from a series of simple operations from Möbius scales ^{[ twelfth ]} .

D. Walba and his colleagues synthesized the first of Möbius -shaped molecular -shaped molecular structures ^{[ 13 ]} , and since this structure has been the subject of interest in chemistry and stereochemistry ^{[ 14 ]} , and in particular in relation to the scale form of DNA molecules. By keeping this application in mind, Erica Flapan studies ^{[ 15 ]} Mathematical symmetries of diving (in) Möbius scales in

${displaystyle mathbb {R} ^{3}}$

.

Möbius scales have also been used for the form of superconductive rings in experiments consisting in studying the spatial structure of drivers in interactions between electrons ^{[ 16 ] , [ 17 ]} .

Finally, they were also used in computer science, as part of linear optimization approaches in whole numbers of set packing and scheduling problems. The scales of Möbius can be used to define the facets of the polytope which describes a continuous relaxation of the problem; These facets are called Möbius scale constraints ^{[ 18 ] , [ 19 ] , [ 20 ] , [ 21 ] , [ 22 ]} .

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