Toroidal Crossing Number
The toroidal crossing number 
of a graph 
is the minimum number of crossings with which 
can be drawn on a torus.
A planar graph has toroidal crossing number 0, and a nonplanar graph with toroidal crossing number 0 is called a toroidal graph. A nonplanar graph with toroidal crossing number 0 has graph genus 1 since it can be embedded on a torus (but not in the plane) with no crossings.
A graph having graph crossing number or rectilinear crossing number less than 2 has toroidal crossing number 0. More generally, a graph that becomes planar after the removal of a single edge (in other words, a graph 
with graph skewness 
) also has toroidal crossing number 0. However, there exist graphs with 
all of whose edge-removed subgraphs are nonplanar, so this condition is sufficient bit not necessary.
If a graph 
on 
edges has toroidal crossing number 
, then 
(Pach and Tóth 2005), where 
denotes the binomial coefficient. Furthermore, if 
is a graph on 
vertices with maximum vertex degree 
which has toroidal crossing number 
, then
 |
(1)
|
where 
is a positive constant (Pach and Tóth 2005).
The toroidal crossing numbers for a complete graph 
for 
, 2, ... are 0, 0, 0, 0, 0, 0, 0, 4, 9, 23, 42, 70, 105, 154, 226, 326, ... (OEIS A014543).
The crossing number of 
on the torus is given by
 |
(2)
|
(Guy and Jenkyns 1969, Ho 2005). The first values for 
, 2, ... are therefore 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, ... (OEIS A008724).
The crossing number of 
on the torus is given by
![nu_t(K_(4,n))=1/2|_n/4_|[n-2(1+|_n/4_|)]](https://mathworld.wolfram.com/images/equations/ToroidalCrossingNumber/NumberedEquation3.svg) |
(3)
|
(Ho 2009). The first values for 
, 2, ... are therefore 0, 0, 0, 0, 2, 4, 6, 8, 12, 16, 20, 24, 30, 36, ... (OEIS A182568). Interestingly, the same result holds for 
, 
, 
, and 
.
The toroidal crossing numbers for a complete bipartite graph 
are summarized in the following table.
 |
1 |
2 |
3 |
4 |
5 |
6 |
| 1 |
0 |
0 |
0 |
0 |
0 |
0 |
| 2 |
|
0 |
0 |
0 |
0 |
0 |
| 3 |
|
|
0 |
0 |
0 |
0 |
| 4 |
|
|
|
0 |
2 |
4 |
| 5 |
|
|
|
|
5 |
8 |
| 6 |
|
|
|
|
|
12 |
REFERENCES
Altshuler, A. "Construction and Enumeration of Regular Maps on the Torus." Disc. Math. 4, 201-217, 1973.
Gardner, M. "Crossing Numbers." Ch. 11 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 133-144, 1986.
Guy, R. K. and Jenkyns, T. "The Toroidal Crossing Number of 
." J. Combin. Th. 6, 235-250, 1969.
Guy, R. K.; Jenkyns, T.; and Schaer, J. "Toroidal Crossing Number of the Complete Graph." J. Combin. Th. 4, 376-390, 1968.
Harary, F. and Palmer, E. M. "A Survey of Graph Enumeration Problems." In A Survey of Combinatorial Theory (Ed. J. N. Srivastava). Amsterdam: North-Holland, pp. 259-275, 1973.
Ho, P. T. "The Crossing Number of 
on the Real Projective Plane." Disc. Math. 304, 23-33, 2005.
Ho, P. T. "The Toroidal Crossing Number of 
." Disc. Math. 309, 3238-3248, 2009.
Pach, J. and Tóth, G. "Thirteen Problems on Crossing Numbers." Geocombin. 9, 195-207, 2000.
Pach, J. and Tóth, G. "Crossing Number of Toroidal Graphs." In International Symposium on Graph Drawing (Ed. P. Healy and N. S. Nikolov). Berlin, Heidelberg: Springer-Verlag: pp. 334-342, 2005.
Riskin, A. "On the Nonembeddability and Crossing Numbers of Some Toroidal Graphs on the Klein Bottle." Disc. Math. 234, 77-88, 2001.
Sloane, N. J. A. Sequences A008724, A014543, and A182568 in "The On-Line Encyclopedia of Integer Sequences."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.
الاكثر قراءة في نظرية البيان
اخر الاخبار
اخبار العتبة العباسية المقدسة
