Klein Bottle Crossing Number

The Klein bottle crossing number of a graph  is the minimum number of crossings possible when embedding  on a Klein bottle (cf. Garnder 1986, pp. 137-138). While the notation is not standardized, Riskin (2001) denotes the Klein bottle crossing number of  as .

The best known example of a graph with nonzero Klein bottle crossing number is the complete graph , which can be embedded on a torus (i.e., it has toroidal crossing number 0) but not on a Klein bottle (Franklin 1934, Riskin 2001).

While a complete list of obstructions for embedding graphs into the Klein bottle is not known as of 2022, Mohar and Škoda (2020) obtained the complete list of 668 obstructions having connectivity 2. The total number of obstructions for the Klein bottle is expected to be in tens of thousands, and possibly even more than a million (Mohar and Škoda 2020).

Riskin (2001) showed that toroidal polyhedral maps with four or more disjoint homotopic noncontractible circuits are not embeddable on the projective plane and that toroidal polyhedral maps with five or more disjoint homotopic noncontractible circuits are not embeddable on the Klein bottle.

Riskin (2001) also gave the Klein bottle crossing numbers of the torus grid graphs  with  for , 4, 5, 6 are 1, 2, 4, and 6, respectively.

REFERENCES

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Franklin, P. "A Six Colour Problem." J. Math. Phys. 13, 363-369, 1934.

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Kawarabayashi, K.-I.; Král', D.; Kynľ, J.; and Lidický, B. "6-Critical Graphs on the Klein Bottle." SIAM J. Discr. Math. 23, 372-383, 2008/2009.

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Lawrencenko, S. and Negami, S. "Irreducible Triangulations of the Klein Bottle." J. Combin. Theory Ser. B 70, 265-291, 1997.

Lawrencenko, S. and Negami, S. "Constructing the Graphs That Triangulate Both the Torus and the Klein Bottle." J. Combin. Theory Ser. B 77, 211-2218, 1999.

Mohar, B. and Škoda, P. "Excluded Minors for the Klein Bottle I. Low Connectivity Case." 1 Feb 2020.

 https://arxiv.org/abs/2002.00258.Riskin, A. "On the Nonembeddability and Crossing Numbers of Some Toroidal Graphs on the Klein Bottle." Disc. Math. 234, 77-88, 2001.

Thomassen, C. "Tilings of the Torus and the Klein Bottle and Vertex-Transitive Graphs on a Fixed Surface." Trans. Amer. Math. Soc. 323, 605-635, 1991.