The Klein bottle crossing number of a graph 
is the minimum number of crossings possible when embedding
on a Klein bottle (cf. Gardner 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).
The complete graph 
has Klein bottle crossing number 0 for 
. Koman (1969) proved that the values for 
, 
, and 
are 1, 4, and 9, respectively.
For complete bipartite graphs, Richter and Širáň (1996) proved that the Klein bottle crossing number
of 
is 
,
where 
is the floor function. Koman (1975) also determined
the exact values for 
with 
, as well as for 
, 
, and 
.
Every Möbius ladder embeds in the projective plane, and therefore also in the Klein bottle, so every Möbius ladder has Klein bottle crossing number 0.
Beaudou et al. (2010) proved that if 
and 
are connected graphs, then
every crossing-minimal drawing of their graph
disjoint union 
on a Klein bottle
has no crossing between a graph edge of 
and a graph edge of 
. Writing 
, 
, and 
for crossing numbers on the Klein
bottle, projective plane, and plane,
respectively, it follows that
 |
Together with the exact values for complete graphs, this gives 
,
,
and 
.
The theorem and formula concern exactly two connected
components. They do not assert an analogous result for a graph
disjoint union of three or more connected
components.
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. Juarez and Salazar
(2003) proved that, for every fixed 
, this crossing number equals 
for all sufficiently large 
, where 
is the floor function.
Their theorem does not give a general explicit threshold for 
. In particular, the Klein bottle crossing numbers of the diagonal
family 
remain undetermined for 
.
on the Klein Bottle." In Recent Advances in
Graph Theory. Prague: Academia, pp. 327-334, 1975.Koman, M.
"New Upper Bounds for the Crossing Number of 
on the Klein Bottle." Časopis Pest. Mat. 103,
282-288, 1978.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. 
in a Surface." J. Graph Th. 21, 51-54,
1996.