Mixing
Problems with proof concerning the number of crossings calculated by nested integrals
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apologies for the horrendous title.
The paper I am talking about can be found here:
https://www.math.nyu.edu/~pach/publications/fewcrossing.pdf
Remark 3.2 explains the construction of "dense" lower bound examples for $k$-planar graphs. The idea is to evenly space out $v$ points on a $sqrtv times sqrtv $ grid and draw edges between vertices that have distance at most $d$, where $d = sqrt frac2epi v$, where $v$ and $e$ denote the number of vertices and edges.
They then derive that the number of crossings is $frac2 pi27 vd^6 - (1 + o(1))$
The equation can be found at the bottom of page 8.
While I understand the intuition of the integrals, I have no idea how they derive the result from it.
graph-theory definite-integrals
asked Jul 22 at 20:10
MrLemming
313
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up vote
0
down vote
favorite
apologies for the horrendous title.
The paper I am talking about can be found here:
https://www.math.nyu.edu/~pach/publications/fewcrossing.pdf
Remark 3.2 explains the construction of "dense" lower bound examples for $k$-planar graphs. The idea is to evenly space out $v$ points on a $sqrtv times sqrtv $ grid and draw edges between vertices that have distance at most $d$, where $d = sqrt frac2epi v$, where $v$ and $e$ denote the number of vertices and edges.
They then derive that the number of crossings is $frac2 pi27 vd^6 - (1 + o(1))$
The equation can be found at the bottom of page 8.
While I understand the intuition of the integrals, I have no idea how they derive the result from it.
graph-theory definite-integrals
asked Jul 22 at 20:10
MrLemming
313
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
apologies for the horrendous title.
The paper I am talking about can be found here:
https://www.math.nyu.edu/~pach/publications/fewcrossing.pdf
Remark 3.2 explains the construction of "dense" lower bound examples for $k$-planar graphs. The idea is to evenly space out $v$ points on a $sqrtv times sqrtv $ grid and draw edges between vertices that have distance at most $d$, where $d = sqrt frac2epi v$, where $v$ and $e$ denote the number of vertices and edges.
They then derive that the number of crossings is $frac2 pi27 vd^6 - (1 + o(1))$
The equation can be found at the bottom of page 8.
While I understand the intuition of the integrals, I have no idea how they derive the result from it.
graph-theory definite-integrals
asked Jul 22 at 20:10
MrLemming
313
apologies for the horrendous title.
The paper I am talking about can be found here:
https://www.math.nyu.edu/~pach/publications/fewcrossing.pdf
Remark 3.2 explains the construction of "dense" lower bound examples for $k$-planar graphs. The idea is to evenly space out $v$ points on a $sqrtv times sqrtv $ grid and draw edges between vertices that have distance at most $d$, where $d = sqrt frac2epi v$, where $v$ and $e$ denote the number of vertices and edges.
They then derive that the number of crossings is $frac2 pi27 vd^6 - (1 + o(1))$
The equation can be found at the bottom of page 8.
While I understand the intuition of the integrals, I have no idea how they derive the result from it.
graph-theory definite-integrals
asked Jul 22 at 20:10
MrLemming
313
asked Jul 22 at 20:10
MrLemming
313
asked Jul 22 at 20:10
MrLemming
313
asked Jul 22 at 20:10
MrLemming
313
313
add a comment |Â
add a comment |Â
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