Mixing
Minimum genus of torus necessary to embed complete graph $K_n$
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You can embed complete graphs $K_1$, $K_2$, $K_3$, and $K_4$ on a genus $0$ torus (a sphere).
The minimal genus of a torus on which you can embed $K_5$, $K_6$, and $K_7$ is a $1$.
Then you need a torus of genus $2$ to embed ...
Is there a formula for any $K_n$, stating the minimum genus necessary to embed $K_n$ on a torus with that genus?
topological-graph-theory
edited Jan 7 at 9:10
Saad
19.7k92352
asked Mar 14 '18 at 16:41
J.ShootJ.Shoot
83
$endgroup$
add a comment |
$begingroup$
You can embed complete graphs $K_1$, $K_2$, $K_3$, and $K_4$ on a genus $0$ torus (a sphere).
The minimal genus of a torus on which you can embed $K_5$, $K_6$, and $K_7$ is a $1$.
Then you need a torus of genus $2$ to embed ...
Is there a formula for any $K_n$, stating the minimum genus necessary to embed $K_n$ on a torus with that genus?
topological-graph-theory
edited Jan 7 at 9:10
Saad
19.7k92352
asked Mar 14 '18 at 16:41
J.ShootJ.Shoot
83
$endgroup$
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
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– GNUSupporter 8964民主女神 地下教會
Mar 14 '18 at 16:43
add a comment |
$begingroup$
You can embed complete graphs $K_1$, $K_2$, $K_3$, and $K_4$ on a genus $0$ torus (a sphere).
The minimal genus of a torus on which you can embed $K_5$, $K_6$, and $K_7$ is a $1$.
Then you need a torus of genus $2$ to embed ...
Is there a formula for any $K_n$, stating the minimum genus necessary to embed $K_n$ on a torus with that genus?
topological-graph-theory
edited Jan 7 at 9:10
Saad
19.7k92352
asked Mar 14 '18 at 16:41
J.ShootJ.Shoot
83
$endgroup$
You can embed complete graphs $K_1$, $K_2$, $K_3$, and $K_4$ on a genus $0$ torus (a sphere).
The minimal genus of a torus on which you can embed $K_5$, $K_6$, and $K_7$ is a $1$.
Then you need a torus of genus $2$ to embed ...
Is there a formula for any $K_n$, stating the minimum genus necessary to embed $K_n$ on a torus with that genus?
topological-graph-theory
topological-graph-theory
edited Jan 7 at 9:10
Saad
19.7k92352
asked Mar 14 '18 at 16:41
J.ShootJ.Shoot
83
edited Jan 7 at 9:10
Saad
19.7k92352
asked Mar 14 '18 at 16:41
J.ShootJ.Shoot
83
edited Jan 7 at 9:10
Saad
19.7k92352
edited Jan 7 at 9:10
Saad
19.7k92352
edited Jan 7 at 9:10
Saad
19.7k92352
19.7k92352
asked Mar 14 '18 at 16:41
J.ShootJ.Shoot
83
asked Mar 14 '18 at 16:41
J.ShootJ.Shoot
83
asked Mar 14 '18 at 16:41
J.ShootJ.Shoot
83
83
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
Mar 14 '18 at 16:43
add a comment |
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
Mar 14 '18 at 16:43
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
Mar 14 '18 at 16:43
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
Mar 14 '18 at 16:43
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Found it. All I needed to search for was "genus of complete graphs" :
http://mathworld.wolfram.com/GraphGenus.html
Ringel and Youngs 1968; Harary 1994, p. 118 :
$$mbox{ceil}left((n-3)(n-4)over 12right)$$
Please do edit the formula with MathJax.
edited Mar 16 '18 at 15:18
user36196
41337
answered Mar 16 '18 at 14:57
J.ShootJ.Shoot
83
$endgroup$
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
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active
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active
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$begingroup$
Found it. All I needed to search for was "genus of complete graphs" :
http://mathworld.wolfram.com/GraphGenus.html
Ringel and Youngs 1968; Harary 1994, p. 118 :
$$mbox{ceil}left((n-3)(n-4)over 12right)$$
Please do edit the formula with MathJax.
edited Mar 16 '18 at 15:18
user36196
41337
answered Mar 16 '18 at 14:57
J.ShootJ.Shoot
83
$endgroup$
add a comment |
$begingroup$
Found it. All I needed to search for was "genus of complete graphs" :
http://mathworld.wolfram.com/GraphGenus.html
Ringel and Youngs 1968; Harary 1994, p. 118 :
$$mbox{ceil}left((n-3)(n-4)over 12right)$$
Please do edit the formula with MathJax.
edited Mar 16 '18 at 15:18
user36196
41337
answered Mar 16 '18 at 14:57
J.ShootJ.Shoot
83
$endgroup$
add a comment |
$begingroup$
Found it. All I needed to search for was "genus of complete graphs" :
http://mathworld.wolfram.com/GraphGenus.html
Ringel and Youngs 1968; Harary 1994, p. 118 :
$$mbox{ceil}left((n-3)(n-4)over 12right)$$
Please do edit the formula with MathJax.
edited Mar 16 '18 at 15:18
user36196
41337
answered Mar 16 '18 at 14:57
J.ShootJ.Shoot
83
$endgroup$
Found it. All I needed to search for was "genus of complete graphs" :
http://mathworld.wolfram.com/GraphGenus.html
Ringel and Youngs 1968; Harary 1994, p. 118 :
$$mbox{ceil}left((n-3)(n-4)over 12right)$$
Please do edit the formula with MathJax.
edited Mar 16 '18 at 15:18
user36196
41337
answered Mar 16 '18 at 14:57
J.ShootJ.Shoot
83
edited Mar 16 '18 at 15:18
user36196
41337
edited Mar 16 '18 at 15:18
user36196
41337
edited Mar 16 '18 at 15:18
user36196
41337
41337
answered Mar 16 '18 at 14:57
J.ShootJ.Shoot
83
answered Mar 16 '18 at 14:57
J.ShootJ.Shoot
83
answered Mar 16 '18 at 14:57
J.ShootJ.Shoot
83
83
add a comment |
add a comment |
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Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
Mar 14 '18 at 16:43