Drawing Planar Graphs Via Dessins D'enfants

Drawing Planar Graphs via Dessins d’Enfants

Kevin Bowman, Sheena Chandra, Anji Li and Amanda Llewellyn

PRiME 2013: Purdue Research in Mathematics Experience, Purdue University

July 21, 2013

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 1 / 25 Belyi Maps

Deﬁnition p(z) 1 1 A rational function β(z) = q(z) , where β : P (C) → P (C), is a Belyi Map if it has at most three critical values, say {ω(0), ω(1), ω(∞)}.

Remarks: 1 P (C) refers to the complex projective line, that is the set C ∪ {∞}. 1 An ω ∈ P (C) is a critical value of β(z) if β(z) = ω for some β0(z) = 0.

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 2 / 25 Belyi Maps

Question Is β(z) = 4z5(1 − z5) a Belyi Map?

1 Form a polynomial equation

ω1 5 5 β(z) = ⇐⇒ ω0 4z (1 − z ) − ω1 = 0 ω0 2 Compute the discriminant 5 5 4 9 5 disc ω0 4z (1 − z ) − ω1 = (constant) ω1 ω0 (ω1 − ω0)

3 Find the roots ω1 1 4 9 5 = ∈ P (C) ω1 ω0 (ω1 − ω0) = 0 = {0, 1, ∞} ω0

Answer Yes, β(z) is a Belyi Map with critical values {0, 1, ∞}

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 3 / 25 Dessin d’Enfant

Deﬁnition Given a Belyi map β(z) = p(z)/q(z) we consider the preimages

B = “black” vertices = β−1(0) W = “white” vertices = β−1(1) E = edges = β−1([0, 1]) F = midpoints of faces = β−1(∞)

We deﬁne the bipartite graph ∆β = (B ∪ W , E) as the Dessin d’Enfant.

Remarks: A bipartite graph is a collection of vertices and edges where the vertices are placed into two disjoint sets, none of whose elements are adjacent Following Grothendieck, “Dessin d’Enfants” is French for “Children’s Drawings” Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 4 / 25 Dessin d’Enfant

Example

β(z) = 4z5(1 − z5)

We found that β is a Belyi map with critical values 0, 1, ∞. We consider its preimages

B = β−1(0) = {0} ∪ {5th roots of 1} W = β−1(1) = {5th roots of 1/2} F = β−1(∞) = {∞}

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 5 / 25 Research Question

Motivating Question Let Γ be a connected planar graph. Can we ﬁnd a Belyi map β(z) such that Γ is the Dessin d’Enfant of this map?

Remarks: A graph is connected if there is a path between any two points A graph is planar if it can be drawn without any edges crossing Any planar graph can be seen as a bipartite graph if we label all vertices as “black” and label all midpoints of edges as “white”

Specific Question Given a specific web or a tree, can we explicitly find its corresponding Belyi map?

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 6 / 25 Methodology

From a graph to a bipartite graph:

1) Label graph’s vertices as “black” 2) Add “white” vertices as midpoints of edges

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 7 / 25 Methodology

From a bipartite graph to a Belyi map: 3) Label ”black” vertices as 1 B = {b1, b2, ..., br } ⊆ P (C)

such that each point bk has ek edges incident.

Since we want B = β−1(0), we must have r Y p(z) p(z) = (constant) (z − b )ek where β(z) = k q(z) k=1 Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 8 / 25 Methodology

4) Label “white” vertices as 1 W = {w1, w2, ..., wn} ⊆ P (C)

such that each point wk has 2 edges incident. Since we want W = β−1(1), we must have n Y p(z) p(z) − q(z) = (constant) (z − w )2 where β(z) = k q(z) k=1 5) Label midpoints of faces vertices as 1 F = {f1, f2, ..., fs } ⊆ P (C)

such that each point fk has dk edges that enclose it. Since we want F = β−1(∞), we must have s Y dk q(z) = (constant) (z − fk ) k=1

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 9 / 25 Methodology

Proposition

If there exist constants bk , wk , fk , p0, q0, r0 ∈ C such that r n s Y ek Y 2 Y dk p0 (z − bk ) − q0 (z − wk ) − r0 (z − fk ) = 0 k=1 k=1 k=1 | {z } | {z } | {z } vertices edges faces

for all z, then the rational function

r p Q (z − b )ek β(z) = − 0 k=1 k Qs dk r0 k=1(z − fk ) is a Belyi map of degree

r s X X 2 n = ek = dk . k=1 k=1

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 10 / 25 List of Results

Cycles Dipoles Prisms Bipyramids Webs Antiprisms Trapezohedrons Wheels Gyroelongated Bipyramid Truncated Trapezohedron Paths Trees Stars

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 11 / 25 Graphing: Mathematica Notebook

http://www.math.purdue.edu/∼egoins/site//Dessins%20d’Enfants.html

Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 12 / 25 Cycles

(zn − 1)2 β(z) = − 4zn