Topological Graph Theory

A Survey

Dan Archdeacon

Dept of Math and Stat

University of Vermont

Burlington VT USA

email danarchdeaconuvmedu

Introduction

Graphs can b e represented in many dierent ways by lists of edges by

incidence relations by adjacency matrices and by other similar structures

These representations are well suited to computer algorithms Historically

however graphs are geometric ob jects The vertices are p oints in space and

the edges are line segments joining select pairs of these p oints For example

the p oints may b e the vertices and edges of a p olyhedron Or they may b e the

intersections and trac routes of a map More recently they can represent

computer pro cessors and communication channels These pictures of graphs

are visually app ealing and can convey structural information easily They

reect graph theorys childhoo d in the slums of top ology

Topological graph theory deals with ways to represent the geometric real

ization of graphs Typically this involves starting with a graph and depicting

it on various types of drawing b oards space the plane surfaces b o oks

etc The eld uses top ology to study graphs For example planar graphs

have many sp ecial prop erties The eld also uses graphs to study top ology

For example the graph theoretic pro ofs of the Jordan Curve Theorem or

the theory of voltage graphs depicting branched coverings of surfaces pro

vide an intuitively app ealing and easily checked combinatorial interpretation

of subtle top ological concepts

In this pap er we give a survey of the topics and results in top ological

graph theory We oer neither breadth as there are numerous areas left

unexamined nor depth as no area is completely explored Nevertheless we

do oer some of the favorite topics of the author and attempt to place them

in context

We b egin with some background material in Section Section covers

map colorings and Section contains other classical results Section exam

ines several variations on the basic theme including dierent drawing b oards

and restrictions Section lo oks at lo cally planar embeddings on surfaces

Chapter gives a brief introduction to graph minors Chapter to random

top ological graph theory and Chapter to symmetrical maps Chapter

contains some op en problems and Chapter is the conclusion

Background Material

In this section we introduce some of the basic terms and concepts of top olog

ical graph theory The reader seeking additional graphtheoretic denitions

should consult the b o ok by Bondy and Murty A more detailed treat

ment of embeddings is in the b o ok by Gross and Tucker We examine

in turn the basic terms surfaces Eulers formula and its consequences the

maximum and minimum genus combinatorial descriptions of embeddings

and partial orders

Basic Terms

A graph G is a nite collection of vertices and edges Each edge has two

vertices as ends An edge with b oth endp oints the same is called a loop

Two edges with the same pair of endp oints are parallel In some applications

it is common to require that graphs are simple that is have no lo ops or

parallel edges In top ological graph theory it is common to allow b oth

Each graph G corresp onds to a top ological space called the geometric

realization In this space the vertices are distinct p oints and the edges are

subspaces homeomorphic to joining their ends Two edges meet only at

their common endp oints An embedding of G into some top ological space X

is a homeomorphism b etween the geometric realization of G and a subspace

of X For convenience we freely confuse a vertex in the graph the p oint in

its geometric realization and the corresp onding p oint when embedded in X

Where should we embed a graph Perhaps the most natural space to

2 2

consider is the real plane R A graph embedded in the plane G R is

called a plane graph a graph admitting such an embedding is planar In a

connected plane graph each comp onent of R G is homeomorphic to an

op en cell However as shown by an embedding of the graph with a single

vertex and two lo ops in the plane it may b e that the closure of this op en

cell is not a closed disk Instead there may b e rep eated p oints along the

b oundary

Surfaces

As we will show not every graph embeds in the plane How then can we

picture it Keeping the space lo cally planar we can try to embed graphs

in surfaces that is compact Hausdorf top ological spaces which are lo cally

homeomorphic to R There are two ways to construct such surfaces take

a sphere and attach n handles or take a sphere and attach m crosscaps We

denote these surfaces by S and S resp ectively By a theorem of Brahana

n m

any surface falls in one of these two innite classes see for details

In particular the surface obtained by adding in n handles and m crosscaps

m is homeomorphic to S A surface S is orientable that is it is

2n+m n

p ossible to assign a lo cal sense of clo ckwise and anticlockwise so that along

any path b etween any two p oints in the surface the lo cal sense is consistent

However S is nonorientable a consistent assignment of sense is imp ossible

It is easily shown that any graph embeds in some surface draw it in

the plane with crossings and use a handle to jump over each crossing

We wish the graph to carry a reasonable amount of information ab out the

surface in which its embedded In particular if the surface has a handle or

crosscap then we want the graph to use that feature For example a single

lo op embedded in a small lo cal neighborho o d of a p oint in a torus do es not

use the handle An embedding is cellular if each comp onent of X G ie

each face is homeomorphic to an op en cell In a cellular embedding any

curve in the surface is homotopic to a walk in the graph Note that only

connected graphs have cellular embeddings Henceforth we declare that all

graphs are connected and all embeddings are cellular If an embedding has

the additional prop erty that the closure of each face is homeomorphic to a

closed disk then the embedding is circular or closed cell CTC

Given an embedded primal graph there is a natural way to form an em

b edded geometric dual graph We place a vertex of the dual in the interior of

each face of the primal embedding Whenever two faces of the primal share a

common edge add an edge of the dual from the middle of one face through

the middle of the common edge to the middle of the other face This dual is

embedded in the surface in a natural manner The duality op erator swaps the

dimensional p oints with the dimensional faces leaving the dimensional

edges xed Observe that the dual of the dual is the embedded primal

graph

A cycle C in a surface S may b e contractible that is homotopic to a

p oint A noncontractible cycle is called essential An essential cycle may

still b e separating that is S C may b e disconnected Noncontractible

separating cycles are homologically but not homotopically null

The Euler Characteristic

Let G b e a graph cellularly embedded in a surface S Supp ose that V is

the number of vertices of G E is the number of edges and F is the num

b er of faces in the embedding The Euler Characteristic of the embedding is

G V E F It is well known that the Euler Characteristic

of the embedding dep ends only on the surface and not on the embedding

If the surface is the sphere with n handles attached then the Euler Char

acteristic is n and if it is the sphere with m crosscaps then the Euler

Characteristic is m We call the quantity the Euler genus of the

surface This parameter has also b een called the generalized genus and the

complexity of the surface Each handle contributes two to the Euler genus

and each crosscap contributes one

Eulers formula can b e used in combination with other inequalities to

derive some interesting b ounds We b egin with the observation that an

embedding of a connected graph which is not a tree has the length of each

face b ounded b elow by the girth g Since the sum of the face lengths is E

this gives g F E In combination with Eulers formula this gives

E V g g

Roughly sp eaking for girth g and xed V each crosscap increasing

by one can carry up to three edges and each handle increasing by two

can carry six edges When g these numbers drop to two edges and four

edges resp ectively

Inequalities of this type are used to show the nonexistence of embeddings

For example supp ose by way of contradiction that K has a planar embed

ding Using Euler genus for the sphere and girth g V

and E for K we violate the preceding inequality This contradiction

shows no such planar embedding exists A similar argument works for K

The Maximum and Minimum Genus

A graph can have many p ossible embeddings on many dierent surfaces

Naturally the extremal embeddings are of interest Dene the minimal

orientable genus of G G to b e the smallest n such that G embeds

on the sphere with n handles Likewise dene the nonorientable genus

G as the smallest m such that G embeds on the sphere with m cross

caps We consider a planar graph to b e of nonorientable genus zero al

though some authors say it is of nonorientable genus one The Euler genus

G minf G Gg Dene the maximum genus G the maxi

mum nonorientable genus G and the maximum Euler genus G

M M

in a similar manner

The maximum and minimum genus completely determine the orientable

surfaces on which a connected graph cellularly embeds This follows from

the interpolation theorem of Duke which states that if a graph embeds

on a sphere with n handles and on one with m handles then it embeds on

all intermediate surfaces The pro of uses the concept of rotations dened in

the following section and the observation that moving a single edge end to a

dierent lo cation in a rotation changes the genus of the resulting embedding

by at most one A similar interpolation theorem for nonorientable surfaces

is due to Stahl

Bounds on these maximum and minimum orientable and nonorientable

genus are given in the following lemma

Lemma Let G be a graph with V vertices E edges and girth g

Then

g E g V G G E V

g E g V G G E V and

G G

The two lower b ounds are proved using Eulers formula with an upp er

b ound on the number of faces similar to the argument that K is nonplanar

The two upp er b ounds are also proved using Eulers formula and a lower

b ound of one face Note that in the orientable case the number of faces is

determined by the graph up to parity so that the lower b ound is either one

or two faces Finally note that in the nonorientable case there is always an

embedding with just a single face I usually cite but I have also heard

the result credited to Edmonds We examine the maximum genus parameter

more closely in Section

The third inequality shows that the nonorientable genus cannot b e to o

large compared to the orientable genus However conversely there are graphs

of nonorientable genus one and orientable genus n so there is no such

b ound in the other direction

Graphs in which equality holds in the third equation are called orientably

simple For example K embeds in the torus but not in Kleins b ottle and

so is orientably simple To the authors knowledge there is no detailed study

of orientably simple graphs

Combinatorial Descriptions of Embeddings

We need a convenient combinatorial way to describ e an embedding It is

easiest to b egin with an orientable surface The following was implicit in the

work of Heter with Edmonds and Youngs usually credited

with b eing the rst to resp ectively dualize and formalize the pro cess

Fix a consistent orientation at each p oint on the surface say anticlock

wise By lo oking at the a neighborho o d of a p oint this orientation determines

a cyclic p ermutation of the edges with ends at a vertex v or more precisely in

the case of lo ops of the edge ends at v We call such a cyclic p ermutation a

local rotation at v A rotation on a graph G is a collection of lo cal rotations

one at each vertex Some authors prefer the term rotation scheme As we

have shown an oriented embedding determines a rotation

Conversely supp ose that we are given a rotation We will show how

to construct an embedding into an orientable surface which determines this

particular rotation First use the rotation to trace out the facial walks An

arc is dened by xing one of the two p ossible directions on an edge in a

graph Begin by walking along an arc in a graph with a rotation Up on

reaching the other end of the arc at the vertex v the lo cal rotation at v

leads us to another edge end Continue the walk along the arc on that edge

ro oted at v Pro ceeding in this manner trace out a walk in the graph This

Figure A rotation embedding K in the torus

walk traverses each directed edge at most once and is indep endent of the

starting arc Doing this for each p ossible arc determine a set F of walks

which traverse each arc exactly once Next for each f F of length n take

a convex ngon together with its interior in the plane and lab el the sides

as in the walk of length n This will serve as one face in the embedding

Finally glue the F p olygons together using the lab eling determined by the

graph The result is a surface since each edge lies in two walks and around a

vertex the p olygons line up in the cyclic ordering given by the lo cal rotation

Moreover this surface is orientable and so is homeomorphic to some S

We illustrate this with an example Let K b e the complete graph on

the vertex set the integers mo dulo ve Around vertex we lab el the edges

dep ending on their other endp oint The lo cal rotation at vertex is

The lo cal rotations at vertices and are given in Figure

Tracing the faces of this embedding yields ve quadrilaterals the cyclic

symmetry helps simplify the calculation These t together to form a torus

as shown in Figure In this gure the top of the rectangle is identied with

the b ottom and the left with the right to recover the torus

There are two p ossible ways to consistently orient a surface clo ckwise

and anticlockwise Each graph embedded on the surface will lead to exactly

two dierent rotations dep ending on the sense of the lo cal rotations That

is the rotations are in corresp ondence with embeddings of the graph into

oriented surfaces and in corresp ondence with embeddings into orientable

surfaces There are exactly deg v rotations so this is the number

v V G

of dierent cellular embeddings of G into oriented surfaces

How do we describ e a nonorientable embedding We use a combinatorial

structure called a signed rotation This consists of a rotation and a signature

an assignment of a plus or minus on each edge

We rst describ e how to get a signed rotation from an embedded graph

Let G b e embedded on a p ossibly nonorientable surface Fix a lo cal ori

entation at each vertex If the surface is nonorientable this cannot b e done

consistently As b efore this lo cal orientation determines a lo cal rotation at

each vertex To get the signature lab el an edge with a plus if the two ori

entations at the ends agree and lab el it with a minus if they disagree This

gives a signed rotation

Conversely we can take a signed rotation and construct an embedding

As b efore we use the signed rotation to rst trace out the facial walks This

time we keep track of the current state of a walk either plus or minus We

start out at one end of an edge in a plus state We walk along that edge If

the edge is plus we keep the current state if it is minus we toggle the current

state When we reach the other end if our current state is plus we use the

lo cal rotation if our current state is minus we use the inverse of the lo cal

rotation We continue this way until we return to the same edgeend in the

same state This algorithm traces out walks which contain every edge exactly

twice The toggling b etween states when we traverse along a negative edge

corresp onds to the fact that the lo cal rotations disagree at the ends Instead

of using the lo cal sense we last used we must use the opp osite sense Once

we have traced the facial walks we identify them with the edge of ngons as

b efore and use the lab elings to reconstruct the surface

We illustrate this pro cedure with an example embedding K in the pro

jective plane We take as the vertices of K the integers mo dulo again

identifying the edges incident with a vertex by their other endp oint The

lo cal rotation at is the other lo cal rotations are given in Figure

The edges signed minus are and The resulting embedding is depicted

in Figure There the pro jective plane is recovered by identifying each p oint

x on the b oundary circle with its antipo dal p oint x

Two dierent signed rotations may lead to the same embedding For

while si example we can switch the lo cal rotation to the inverse

multaneously toggling the sign on each edge incident with v The resulting

negative edges

p ositive edges

Figure A signed rotation embedding K in the pro jective plane

signed rotation is dierent but the embedding describ ed is the same We

call this a local switch of sense Any two signed rotations leading to the

same nonorientable embedding are related by a sequence of lo cal switches

of sense Hence any embedding can b e describ ed by dierent signed

rotations Counting the number of signed rotations it follows that there are

#E #V

deg v dierent cellular embeddings of G into surfaces

v V

Knowing the total number of embeddings and the number of orientable

embeddings we can calculate the number of nonorientable embeddings These

counts reveal that most embeddings are in nonorientable surfaces An em

b edding describ ed by a signed rotation is in an orientable surface if and only

if it is equivalent under a sequence of lo cal switches to a signed embedding in

which each edge is p ositive As this is unlikely it conrms that the number

of nonorientable embeddings exceed orientable ones

Partial Orders on Graphs and Embeddings

In many cases it is convenient to place a partial order on the set of graphs

or embedded graphs We mention four such orders in particular

The rst is the subgraph ordering H G Observe that if G embeds

in a surface the H do es as well although the latter embedding may not b e

cellular

The second ordering is the top ological ordering A graph H is an ele

mentary subdivision of G if it is formed from G by deleting an edge uv and

replacing it with a path uw v where w is not a vertex of G In this case we say

that G is formed from H by supressing the degree two vertex w Two graphs

H and G are homeomorphic if they are related by a sequence of elementary

sub divisions and supressing degree two vertices The name arises b ecause G

and H are homeomorphic as graphs if and only if they are homeomorphic

as top ological spaces It follows that embedding prop erties are determined

by the homeomorphism class of a graph The topological order is dened by

H G if and only if H is homeomorphic to a subgraph of G Again if G

embeds in a surface then so do es any H G but the latter embedding may

not b e cellular

The third order is the minor order The elementary op erations dening

H G are of three types The rst is the deletion of isolated vertices in G

The second is the deletion of edges H G n e The third is the contraction

of an edge e H Ge dened by rst deleting e and then identifying its

endp oints If H can b e formed from G by a sequence of these op erations

then H is a minor of G The edge deletion and edge contraction of a non

lo op can b e done in a host surface where by convention contracting an

essential lo op is equivalent to deleting it so that if G embeds in a surface

then so do es any minor H Note that for embedded graphs edge contraction

and edge deletion are dual op erations that is contracting an edge in the

primal graph corresp onds to deleting the corresp onding edge in the dual

graph Equivalently deleting a primal edge corresp onds to contracting the

dual edge

The fourth ordering includes the three minor op erations and the Y op

eration In this op eration a degree vertex w adjacent in G with vertices

x y z is deleted and edges xy y z z x are added As with the minor op era

tions the Y op eration can b e done to a graph embedded in the surface

Hence if G embeds on a surface and H G then H also embeds on that

surface

We close by mentioning that o ccasionally one considers the Y op eration

the inverse to Y and the class of graphs equivalent under Y and Y

op erations Finally a slight extension of the last partial order involves rst

sub dividing an edge joining two degree vertices then p erforming a Y

op eration on each of the vertices The resulting subgraph lo oks like a b ow

tie K C

5 4

Map Colorings

In Francis Guthrie was coloring a map of England Each region was

to get a color and when two regions shared a b oundary line they were to

b e colored dierently In a ash of insight he asked what was the fewest

number of colors needed not just for this map but for any map He

conjectured that four colors suce but a correct pro of of this fourcolor

conjecture was many years coming In this section we examine such map

colorings and related problems

A map is an embedded graph A coloring of the map is an assignment

of colors to the faces The coloring is proper if whenever two faces share a

common edge they receive dierent colors Colorings herein will b e prop er

unless otherwise stated A map has a prop er coloring if and only if each edge

lies on the b oundary of two distinct faces A coloring of a map is equivalent

to a vertex coloring of its dual that is assigning a color to each vertex so

that adjacent vertices receive distinct colors

Planar Graphs

Kainen and Saaty write One of the many surprising asp ects of the

fourcolorconjecture is that a number of the most imp ortant contributions

to the sub ject were originally made with the b elief that they were solutions

One of the rst of these was by Kemp e who introduced the recoloring

metho ds now known as Kemp e chains Heawoo d p ointed out the error

in Kemp es argument but was able to mo dify it to give a correct pro of that

every planar graph was colorable Tait introduced the relation with

edgecoloring cubic plane graphs He to o thought he had solved the color

problem his mistake was b elieving that every cubic graph was Hamiltonian

Petersen claried the relation with edgecolorings and introduced his

famous graph see

The FourColor Theorem was proved by App el and Haken in

The pro of was at rst controversial in part b ecause of the reliance on long

computer calculations However the result has b een proven several times

indep endently most recently by Rob ertson Sanders Seymour and Thomas

All pro ofs to date rely on the same basic technique of nding an

unavoidable set of reducible congurations We refer the reader to for

a description of these metho ds

We explore the relation b etween face and edge colorings of plane graphs

rst introduced by Tait A cubic graph is one which every vertex is

incident with exactly three edges A Tait coloring of a cubic graph is a

coloring of the edges such that at each vertex each color app ears exactly

once

If a planar graph is face colored then it can b e edge colored This

can b e seen by using the elements of the Klein group Z Z as face colors

2 2

the edge coloring then assigns each edge the sum of the colors on either side

The converse is also true if a planar cubic graph can b e Tait colored then

it can b e facecolored So the claim that every bridgeless planar cubic

graph can b e Tait colored is equivalent to the claim that every lo opless

planar graph can b e colored

The relationship with the fourcolor conjecture led to a search for cubic

graphs which could not b e Tait colored Since the graphs are bridgeless and

exactly three edges meet at each vertex there do not exist four pairwise

adjacent edges That is there is no immediate obstacle to Tait colorings

However there are several other trivial reasons a graph cannot b e Tait

colored For example a graph with a lo op has no prop er edge colorings

Likewise any graph with a cutedge cannot b e Tait colored To see this

observe that the union of any two color classes in a Tait coloring forms a

subgraph which is regular of degree two Hence every edge is in a cycle and

the graph must b e edgeconnected

In Martin Gardner called nontrivial non Tait colorable graphs

snarks after the mythical creature from a p o em by Lewis Carroll By

nontrivial he meant that a snark must b e cyclically edgeconnected and of

girth at least The idea was to avoid graphs which were not Tait colorable

but which contained easy reductions to smaller non Tait colorable graphs

We refer the reader to Swart and to Issacs for interesting discussions

ab out what constitutes a trivial or p etty snark and should b e exiled

The most famous snark is the Petersen graph discovered in

It was over fty years later b efore another example was found In

Danilo Blanusa found a non Tait colorable graph on vertices the

same techniques lead to a second such graph of the same order In

Blanche Descartes found an example on vertices It was not until

that a fourth example was found by Szekeres At this time several

p owerful construction techniques were developed by Issacs yielding two

innite classes of snarks One of these classes the ower snarks had b een

discovered indep endently three years earlier by Grinberg in some unpublished

work Recently Brinkman p ersonal communication has found all snarks on

or fewer vertices For other recent results see

Four colors suce to vertex color any planar graph but only if they are

the same four colors available at each vertex Supp ose that we have a list of

colors available at each vertex but the lists may b e dierent A list coloring

assigns each vertex one of the colors in its list as usual adjacent vertices

receive dierent colors At rst glance this should b e easier than standard

colorings where all of the lists are the same After all if the lists on the ends

of an edge are dierent then it is more likely that the colors on the ends are

distinct Dene the list chromatic number of G G as the smallest k such

that every assignment of lists of size k to the vertices has a list coloring The

list chromatic number of the plane is the maximum G over all planar G

Theorem The list chromatic number of the plane is ve

Thomassen shows suciency with an elegant pro of that every planar

graph is list colorable Voigt gives an example of a planar graph and

a list assignment of four colors which can not b e list colored

Four colors suce for planar graphs When do three We mention two

famous color theorems In Heawoo d proved that a plane trian

gulation is vertex colorable if and only if every vertex was of even degree

Grotzsch proved that every trianglefree planar graph was vertex

colorable My favorite pro of of this theorem is due to Thomassen

Thomassen gives a list version of Grotzschs Theorem Krol

noted that a plane graph is colorable if and only if it is a subgraph of

an Eulerian triangulation It would b e interesting to nd either a common

extension of or relationship b etween Heawoo d and Grotzschs theorems No

go o d characterization is to b e exp ected since Garey Johnson and Sto ck

meyer showed that in general the problem is NPcomplete We refer the

reader to Steinberg for a recent survey of this three color problem

The color problem is easy A plane graph is colorable if and only

if every face is b ounded by a walk of even length The color

problem is not quite p ointless but it is edgeless

Coloring the faces of a plane graph is equivalent to coloring the vertices

of the dual What if we try to color the vertices and faces simultaneously

Sp ecically a coupled coloring of an embedded G is a coloring of the vertices

and faces so that adjacent or incident elements receive distinct colors Ringel

conjectured that every plane graph has a coupled coloring The the

orem was rst shown for cubic graphs and then for trianglefree graphs

Boro din proved the even stronger result that every graph

which embeds in the plane so that each edge crosses at most one other has

chromatic number at most six

A total coloring of a graph colors vertices and edges so that adjacent

or incident elements receive distinct colors The total chromatic number

G is the minimum number of colors needed It is b ounded b elow by

where is the maximum degree Indep endently Behzad and Vizing

conjectured that for simple graphs it is b ounded ab ove by

This conjecture is known to b e true for several classes of graphs Various

general upp er b ounds are also known We refer the reader to for a

discussion of the known results The conjecture is true for plane graphs

except for the cases The low degree cases are due to Rosenfeld

and Kosto chka These cases do not

use planarity The high degree cases are due to Boro din and

Andersen and do use planarity

An entire coloring of a plane graph colors the vertices faces and edges

simultaneously so that adjacent or incident elements receive distinct colors

Kronk and Mitchem conjectured that the entire chromatic number

of a graph was equal to where is the maximum degree This is true

for plane cubic graphs by M Neub erger as rep orted by Izbicki and

for plane graphs with suciently large by Boro din The

conjecture is still op en for

Along these lines we also mention Vizings Conjecture that

every planar graph with can b e prop erly edgecolored in colors

This conjecture is known to b e true for It is false if extended to

What if you color the faces of an embedded graph so that faces that

share a vertex or edge receive dierent colors In the dual form this requires

a vertex coloring so that around each face no color is rep eated These are

called cyclic colorings by Ore and Plummer The b ound dep ends on the

size of the largest face They showed that the cyclic chromatic number

was at most This was improved to b c by Boro din Sanders and

Zhao Plummer and Toft proved that for connected graphs the

cyclic chromatic number is at most They also give a lower b ound of

b c for the general problem

Finally we mention the problem of coloring the faces of a plane cubic map

so that around each edge each four or fewer faces receive distinct colors

Bouchet Fouquet Jolivet and Riviere showed that this can always b e

done in colors Boro din improved the b ound to Sanders and Zhao

further improved this to The conjectured correct answer is colors

Maps on Other Surfaces

We turn our attention to coloring maps on surfaces Dene the chromatic

number of a surface as the maximum chromatic number of all graphs that

embed on that surface The dual question is to ask for the minimum number

of colors needed to prop erly color the faces of every map on that surface For

example the plane has chromatic number four The following Map Color

Theorem settled this question for all surfaces other than the plane

Theorem The chromatic number of S is b g c for al l

g The chromatic number of S is b h c for al l h

The chromatic number of Kleins bottle is

Surprisingly this was proved b efore the color theorem Also surpris

ingly the easy part of the color theorem is hard in the map color theorem

but the hard part of the color theorem is easy in the map color theorem

We elab orate on these two halves of the pro of in turn

It is easy to show that colors are needed for some planar maps you only

need to draw K in the plane In the map color theorem the required number

of colors is the largest n such that K embeds on the surface However

the dicult part was to show that K or some other nchromatic graph

did in fact embed on the smallest surface allowed by Eulers formula with

the exception of K in Kleins b ottle That task broken into cases by

the residue of n mo dulo and the orientablility or nonorientability of the

surface was completed by Ringel and Youngs in see esp for

the nonorientable case A nice account of the pro of is given in Ringel

Conversely it is hard in the plane to prove that four colors suce for all

maps However on other surfaces except the pro jective plane it is easy to

show that the conjectured number of colors n is sucient One need merely

use an Euler characteristic argument to show that every graph on the surface

has a vertex of degree at most n and use induction

There are variations of coloring graphs in surfaces similar to those men

tioned for the plane These include improp er colorings where the num

b er of adjacent vertices of the same color is b ounded generalizations of

Grotzchs theorem coloring trianglefree graphs simultaneously

coloring vertices and faces and acyclic colorings where

any two color classes induce a forest

We dene the list chromatic number of a surface analogously to the list

chromatic number of the plane Archdeacon and Siran unpublished work

have shown that for any surface other than the plane the list chromatic

number is equal to the chromatic number

The relationship b etween edge colorings and face colorings is not as clear

in other surfaces as in the plane Nevertheless we mention an intriguing

conjecture due to Gr unbaum This conjecture states that every simple

graph that triangulates an orientable surface has an edge coloring such that

each color app ears necessarily exactly once around each triangle In the

dual form this states that if a cubic graph the dual of triangulation is

embedded on an orientable surface such that any two faces share at most a

single edge the dual of simple then the graph can b e Tait colored This

conjecture is strictly stronger than the color theorem b ecause any planar

graph can b e placed in a single face of a triangulation of a surface and the

coloring of that triangulation induces a coloring of the planar part as well

It is equivalent to asserting that any orientable embedding of a nonTait

colorable graph must have two faces sharing more than one edge Note that

the orientable condition is necessary as evidenced by K on the pro jective

plane This embedded graph has dual the Petersen graph which is not Tait

colorable hence K do es not have the desired coloring

Although maps on other surfaces may have large chromatic number it is

the case that if the embedding is lo cally planar for large neighborho o ds of a

vertex then the chromatic number can b e b ounded We investigate this in

Section

We close with the wonderful quote from Tutte The FourColour

Theorem is the tip of the iceb erg the thin end of the wedge and the rst

cuckoo of spring

Classical Results

In this section we state some of the classical results of top ological graph

theory As noted previously the most natural space for depicting graphs

is the plane In Section we state some theorems characterizing planar

graphs Section gives theorems relating to the maximum and minimum

genus of graphs Section studies graphs which embed on a xed surface

Characterizing Planar Graphs

An imp ortant early question is to characterize those graphs which embed

on the plane One such characterization was given by Kuratowski in

The same theorem was proven indep endently and roughly concurrently

by Frink and Smith who never published their pap er after hearing of

Kuratowskis pro of This theorem is very imp ortant and is the most cited

research pap er in graph theory

Theorem A graph is planar if and only if it does not contain a subgraph

homeomorphic to K or to K

5 33

We earlier proved that the two graphs in question were not planar from

which it follows that any graph containing a top ological copy of these graphs

is nonplanar The b eauty of the theorem lies in the pro of that these are the

only two top ologically minimal nonplanar graphs

The theorem can b e rephrased slightly using the minor ordering where

it was rst stated by Wagner

Theorem A graph is planar if and only if it does not contain K or

K as a minor

To state the next theorem we need some algebraic terminology In this

context a cycle in a graph is a set of edges which are incident with each

vertex an even number of times Equivalently it is a subgraph in which each

comp onent is Eulerian A cocycle is a minimal edgecut in the graph Note

that each cycle intersects each co cycle in an even number of edges Consider

subsets of edges as vectors over the integers mo dulo two where addition is

dened as the symmetric dierence of sets Then the collection of cycles is

a subspace Z G as is the collection of co cycles B G These subspaces are

orthogonal

A graph G is an algebraic dual of a graph G if there is a function

E G E G such that C is a cycle of G if and only if C is a co cycle

of G The following is due to Whitney

Theorem A graph is planar if and only if it has an algebraic dual

In fact if G and G are algebraic duals then there exists an plane em

b edding of G so that G is the geometric dual

Let G b e a connected plane graph with face set F Each face b oundary

is a simple closed walk whose edges form a cycle The collection of any F

of these cycles forms a basis for the cycle space Moreover no edge app ears

in more than two members of this collection MacLanes Theorem gives

the converse

Theorem A graph is planar if and only if there is a collection of cycles

which generate the cycle space together with one additional cycle such that

every edge is in exactly two of these cycles

Other authors have given similar algebraic characterizations of

planar graphs

The Genera of Imp ortant Graphs

Recall that the hard part of the Map Color theorem was establishing the min

imum orientable and nonorientable genus of the complete graph K Lower

b ounds on these genera are given by Eulers formula However construct

ing embeddings achieving these b ounds can b e dicult Similarly what

are the minimum genera of complete bipartite graphs regular complete tri

partite or quadripartite graphs the cub es Q and the general o ctahedra

K K nK The answers are given in the following table

n(2) 2n 2

Graph Comments

K dn n e dn n e n K

n 7

K dn m e dn m e n m

K n n

nnn

K n K

4(n) 4(3)

n3 n2

Q n n n

K n n n mo d

n(2)

The results for K form the Map Color Theorem The

orientable and nonorientable genus of complete bipartite graphs were found

by Ringel The orientable genus of complete regular tripartite

graphs was found by Ringel and Youngs and by White for a

particularly nice pro of see Stahl and White The orientable genus of

regular quadripartite graphs is due to Garmen and Jungerman the

sp ecial case K is due to White The orientable genus of the cub e has

4(3)

b een found by several authors The nonorientable genus is due

to Jungerman For low dimensional cub es we note that the formula

for the orientable genus holds for n while Q and Q

4 5

The genus of the o ctahedron is found in Three of these

table entries are question marks To the authors knowledge these values

are unknown although my literature search may have b een incomplete The

genera undoubtedly equal the lower b ound given by Eulers formula except

p ossible for some small values of n

If a graph breaks into pieces along a small set of vertices then it might

b e p ossible to relate the genus of the graph with the genus of the pieces The

rst theorem along these lines is due to Battle Harary Ko dama and Youngs

Theorem The genus of a graph is the sum of the genus of its blocks

The Euler genus is also additive over the blo cks of the graph The nonori

entable genus is not additive counterexamples are the onep oint union of

two orientably simple graphs Stahl and Beineke show that the nonori

entable genus of the onep oint union of two graphs diers from the sum of

their nonorientable genera by at most one

The orientable nonorientable and Euler genus are all nearly additive

over p oint unions that is the genus of a p oint union diers from

the sum of the genera of the comp onents by at most a constant However

these parameters b ehave quite dierently when amalgamating over three or

more vertices Archdeacon has shown that the nonorientable and Euler

genus are b oth almost additive over k p oint unions the constant dep ends on

k but there exist graphs G and G and a p oint union G G such

n n

that G G G G n

n n

The maximum genus of a graph turn out to b e an easier parameter to

calculate than the minimum genus We b egin with the maximum orientable

genus A large genus surface has a small number of faces A graph is upper

embeddable if it has an embedding with or faces the parity is deter

mined by the graphs Betti number This embedding necessarily achieves

the maximal genus Xuong gave a remarkable theorem determining the

maximum genus of a graph To state the theorem let C H denote the

number of comp onents of H with an o dd number of edges

Theorem Let T be the set of al l spanning trees of a graph G Then

G max bE V C G T c

M T T o

The constructive p ortion of Xuongs Theorem is esp ecially nice A is

a pair of edges with a common endp oint Xuong shows that if G has an

embedding with either one or two faces then so do es G The construction

of the desired maximal embedding pro ceeds by rst embedding a spanning

T achieving the ab ove maximum with a single face then successively adding

as many s as p ossible creating a upp er embedded subgraph and nally

adding in the remaining C G T edges each increasing F by one

Xuongs theorem implies that if a graph has two disjoint spanning trees

then it is upp er embeddable By a result of Kundu any edge

connected graph has two disjoint spanning trees Hence any edgeconnected

graph is upp er embeddable These include the complete multipartite graphs

and cub es listed in the table for minimum genus

Xuongs theorem gives an easily applied certicate to verify that a large

genus embedding exists The following theorem of Neb eskys gives

an easily applied certicate to verify that no embedding exists in a higher

surface These two theorems are very p owerful used in concert Let cH

denote the number of comp onents of a graph H and let oH denote the

number of comp onents with o dd Betti number

Theorem The minimum number of faces in an embedding of G is

max fcG A oG A Ag

AE (G)

The maximum nonorientable genus is very easy to calculate Every graph

has an embedding into a nonorientable surface with just a single face This

theorem was rst noted by Edmonds Stahl gave a pro of which

includes the nonorientable version of Dukes interpolation theorem Geomet

rically b egin with a minimum genus embedding of G with two or more faces

Find an edge e lying on two distinct faces Sew a crosscap in the surface in

the middle of this edge Then the resulting embedding has one fewer face

Continue in this way until only one face remains

The maximum orientable genus is not additive over k connected com

p onents but it is nearly additive The maximum nonorientable genus is

additive over all unions

One topic of interest is to nd the smallest p ossible value for the maximum

genus of graphs in a certain class For example the maximum genus of G is

at least G for simplicial graphs and this b ound is tight Chen has

shown that G is a tight lower b ound for connected graphs and

for simplicial connected graphs Chen Archdeacon and Gross give

tight lower b ounds for k connected and for k edgeconnected graphs

We close our discussion on the minimum and maximum genus of a graph

with the computational asp ects

Theorem Determining the minimum orientable genus of a graph is NP

complete Thomassen There is a polynomialtime algorithm to nd

the maximum orientable genus of a graph Furst Gross and McGeoch

The techniques of extend easily to show that determining the mini

mum nonorientable genus is also NPcomplete Since every graph embeds in

a nonorientable surface with a single face determining the maximum nonori

entable genus is trivial Similarly the determining the minimum Euler genus

is NPcomplete while the maximum is trivial

Graphs on a Fixed Surface

To date we have fo cused on the graph used in the embedding namely given

a graph what surfaces do es that graph embed in We now ask a similar

question fo cusing on the surface given a surface what graphs embed on

that surface The rst such characterization of this type was Kuratowskis

Theorem and the closely related Wagners theorem These characterize

planarity by excluding top ological subgraphs or minors Various sources

cite Konig and Erdos for the origin of the conjecture that there are similar

theorems for other surfaces Sp ecically that for each surface S there exists

a nite set I S whose top ological or minor exclusion characterizes embed

ding in S Graphs in I S are called irreducible since they do not embed in

the surface but any prop er subgraph or minor do es embed

The rst result for a surface other than the plane is due to Archdeacon

Theorem There are exactly graphs topological irreducible graphs for

the projective plane

The graphs were originally found by Glover Huneke and Wang Neil

r d

Rob ertson found the graph Archdeacon proved that the list was com

plete Vollmerhaus indep endently veried this completeness unaware

of Archdeacons work These top ological graphs corresp ond to a set of

excluded minors This is implicit in and rst stated explicitly in

The pro jective plane is the only other surface for which a complete list

of irreducible graphs is known The author has p erformed calculations giv

ing hop e that a complete list for the torus may b e assembled using suitable

computer programs Phil Huneke estimates that there may b e top o

logically irreducible graphs

The ErdosKonig niteness conjecture has b een solved in the armative

for all other surfaces

Theorem For each surface there is a nite list of excluded topological

subgraphs which characterize embeddability on that surface Similarly there

is a nite list of excluded minors

Several teams of researchers worked indep endently to establish this result

Archdeacon and Huneke proved the result for nonorientable surfaces

They also proved a similar result for graphs which are minimal with resp ect

to Euler genus Their pro of was constructive providing a metho d in theory

at least for nding the graphs Bo dendiek and Wagner gave a similar result

for orientable surfaces Their pro of was assembled in

Rob ertson and Seymour approached the problem entirely dierently ex

amining a conjecture by Wagner that any innite set of graphs contains one

which is a minor of another This much stronger conjecture implies the

ErdosKonig conjecture since no two irreducible graphs are comparable In

they gave a pro of of this for graphs of b ounded genus which implies

b oth the orientable and nonorientable cases They have since given a pro of of

Wagners conjecture in general Their pro ofs are nonconstructive but

apply in much broader scop e and have many other imp ortant applications

We briey survey their results in Section

We close by noting the plausible conjecture by Glover and indep endently

by Vollmerhaus p ersonal communications that any graph which is irre

ducible for a surface has every edge in a top ological K or K This conjec

5 33

ture was disproved by Brunet Richter and Siran with a clever toroidal

example It is true however for nonorientable surfaces and for connected

graphs

Variations on the Theme

In this section we examine some variations on the classical theories mentioned

ab ove We b egin with restricting our attention to the plane but allowing

edges to cross one another We then examine other planarity restrictions

Next we describ e ways of embedding graphs in which some of the information

either parts of the lo cal rotation or the signature have b een preordained

Finally we consider pseudosurfaces

Drawings in the Plane

Supp ose that we want to depict K in the plane We cannot do it without

edge crossings but we can do it if we allow edges to cross A drawing is like

an embedding of the geometric graph in the surface except that we do not

require the function to b e We put the following restrictions on crossings

in our drawings First all of the vertices must b e distinct p oints Second

the interior of an edge may not pass through any vertex p oint Third any

two edges share at most a nite number of p oints in common

With these restrictions we can depict any graph in the plane for example

place the vertices of K on the vertices of a convex ngon and connect them

Figure Mo difying drawings to reduce crossings

in pairs with straight line segments However we have the cost of having

edge crossings A frugal mathematician will try to nd a drawing of a graph G

with the minimal number of crossings This parameter is called the crossing

number of G denoted cr G

We make several elementary observations ab out crossings First when

two edges meet at a p oint they should cross instead of meeting tangentially

When they meet tangentially then the drawing can b e mo died in a lo cal

manner to reduce the total number of crossings Secondly no pair of edges

with a common endp oint cross For if this o ccurs then the drawing can b e

mo died to remove that crossing and hence decrease the number of crossings

Thirdly no pair of edges can meet in more than one p oint Again if this

o ccurs there is a similar drawing with fewer crossings A drawing satisfying

these restrictions is called good The mo dications transforming any drawing

into a go o d drawing are illustrated in Figure A drawing with the minimum

number of crossings is necessarily go o d Henceforth all drawings will b e go o d

unless otherwise sp ecied

The minimum number of crossings in a drawing of G where all edges

are straight line segments is called the rectilinear crossing numbers cr G

Clearly cr G cr G Equality need not hold in general

What is known ab out the crossing numbers of various classes of graphs

Not much In fact the crossing number of the complete graph K is unknown

in general as is the crossing number of K Another class of interest is

the pro duct of cycles C C Known values are summarized in the table

n m

b elow

Graph cr G cr G Comments

n n1 n2 n3 1

b cb cb cb c cr K n K

n n

4 2 2 2 2

n1 m m1 n

cb cb cb c cr K cr cr K b

nm nm

2 2 2 2

C C nm nm m n

m n

We note that cr K cr K only for n and n In all other

n n

cases strict inequality holds

Most of the b ounds given are upp er b ounds These are in general demon

strated by exhibiting sp ecic drawings with these number of crossings Much

more dicult is the problem of demonstrating lower b ounds of stating that

every drawing must have a particular number of crossings The b ound on

cr K is known to b e exact for n see eg The b ound on

cr K is exact for n and for n m The b ound

on cr C C is exact for n for n and for

n m

n In each case the rst reference or two gives the pro of for

m n while the last reference uses an induction argument to extend this

for general m Note that K is the smallest complete graph for which the

rectilinear crossing number is not known

The problem of nding the crossing number of K is sometimes known

as Turans brickyard problem The idea is that each of n kilns is connected

to each of m shipping centers by rail tracks The carts used to transp ort

the bricks from kiln to shipping center are likely to derail where rails cross

Hence it is convenient to minimize the number of crossings

Can anything b e said ab out the crossing number of complete or complete

bipartite graphs In each case the b ound ab ove is conjectured to b e exact

Perhaps some partial progress could b e made by proving that the b ounds

were asymptotically exact that is by proving eg lim cr K n

In an optimal drawing of K each of the n induced K subgraphs o ccurs

n n1

with at least cr K crossings Accounting for multiplicities this shows

that cr K n is increasing and so this limit exists The b est known b ound

currently is due to Kleitman who shows that the limit is at least

For K the conjectured equality shows that lim cr K n should b e

nn nn

If this limit holds then so do es the one for the complete graphs

For the rectilinear crossing number Jensen has demonstrated an

upp er b ound on cr K n that is asymptotically He has since pri

vate communication reduced this to It is conjectured that the correct

value is also in particular that lim cr K cr K Finally we

n n

note that it is conjectured that the rectilinear crossing number equals the

crossing number for K

We turn our attention to the maximum number of crossings in a drawing

of G cr G This maximum exists since our drawings are go o d so the

number of crossings is b ounded ab ove by the number of pairs of nonadjacent

edges The maximum crossing number is easily calculated for complete and

complete bipartite graphs In particular cr K each induced K

M n 4

can have at most one crossing and choosing the vertices on a circle achieves

n m

this Likewise cr K each induced C can have at most one

M nm 4

2 2

crossing and placing the vertex parts on two parallel lines achieves this

The maximum rectilinear number app ears not to have b een widely studied

except for In general this should b e much less than the maximum

crossing number

When can a graph have a drawing so that every pair of nonadjacent edges

cross John Conway calls a graph thrackled if there exists such a drawing

The curious word also refers to a tangle of shing line which such drawings

often resemble

Conjecture If a connected graph can be thrackled then E V

A connected graph with the same number of vertices as edges necessarily

has a single cycle Such graphs are sometimes called unicycles

Partial results towards the thrackle conjecture are sparse As noted ab ove

the graph cannot have an induced C Woo dall showed that any cycle of

length greater than can b e thrackled Likewise any tree can b e thrackled

If a graph can b e thrackled then so can its subgraphs With some extra

work it can b e shown that the thrackle conjecture reduces to showing that

the onep oint union of two even cycles cannot b e thrackled It is known

that if a graph can b e thrackled then E V

A systematic study of the maximum crossing number cr G is hamp ered

by the fact that the parameter is not known to b e monotone That is if H

is a subgraph of G then is cr H cr G One would exp ect this to b e

M M

true However as shown in a drawing of H do es not always extend to

a drawing of G which negates the obvious way to pro ceed

Various p eople have investigated the crossing number of other surfaces

Kainen worked out a lower b ound using Eulers formula Sp ecically

a graph with girth g on a surface of Euler genus has crossing number at

least E V g g Upp er b ounds for a few graphs

are given by exhibiting drawings

We close by noting that it would b e interesting to nd the total number

of nonisomorphic drawings of a graph However aside form some results of

Harb ourth on small graphs p ersonal communication not much is known in

this area

Thickness

Not every graph is planar What is the smallest n such that G can b e

written as the union of n subgraphs This number is called the thickness

of G denoted G One application of the thickness is when the graph

represents a computer circuit to b e laid out on a stack of circuit b oards The

thickness represents the number of b oards needed

A planar graph has the number of edges b ounded by a function of the

number of vertices This leads to the following Lemma

Lemma For a simple graph G G dE V e Moreover if

G is trianglefree then G dE V e

Can this b ound b e achieved Not in general However it is true for

several interesting classes of graphs The known results are summarized in

the following table

Graph Thickness Comments

K bn c n

K dnmn m e see exceptions b elow

Q bnc see

K dne

n(2)

The thickness of K for n mo d was proven by Beineke and Harary

The remaining congruence class was shown indep endently by Alekseev

and Gonchakon and by Vasak Some small individual cases were

done by hand including n by Mayer

The thickness of K is unknown in the case n m b oth are o dd and

there is an integer k with n k n with m bk n

n k c The smallest such case is K Beineke Harary and Mo on

1929

did the known cases

The thickness of the o ctahedral graphs follows quickly from that of the

complete graphs

Several authors have investigated thickness on other surfaces that is

partitioning the edgeset of graphs into subgraphs of a certain genus We

refer the reader to for a survey of these results

Bo ok Embeddings

Bo oks provide another drawing b oard on which to depict a graph A page is

closed halfplane A book is a collection of pages identied along the b oundary

of the halfplanes This common b oundary is called the spine How many

pages are needed to depict a graph

Theorem Any graph embeds in a book with three pages

The following pro of is due to Babai draw the graph in the plane so that

all crossings involve only two edges and these crossings all lie on a common

line This plane forms two pages of the b o ok Sew the third page along this

line The crossings can easily b e removed using this extra page

A more common form of b o ok embeddings is to require that the vertices

lie along the spine of the b o ok and that edges lie entirely in one page These

restrictions are useful in the applications to VLSI layouts where the pages

can represent circuit b oards or queues used in scheduling tasks

Dene the pagenumber of a graph pnG as the minimum number of

pages need to draw G in this manner

Since two pages form R one might guess that every planar graph has

pagenumber However if this were so then one could always add edges

along the spine so that the given graph was a spanning subgraph of a Hamil

tonian graph This cannot always b e done although it is true for trianglefree

graphs The b est b ound is due to Yannakakis who showed that any

planar graph can b e embedded in a b o ok with pages and that pages were

sometimes necessary

The pagenumber of K is dn e The lower b ound can b e easily seen

since in any ordering of the vertices along the spine there is a set of dn e

edges which cross pairwise and hence must lie on dierent pages The upp er

b ound is established by example

The pagenumber of the complete bipartite graph is surprisingly a much

harder question and is still unknown The b est known b ound is pnK

dn me

We refer the reader to the seminal works by Chung Leighton and Rosen

b erg and Bernhart and Kainen for further discussions on the sub ject

Relative Embeddings

As noted in Section any embedding can b e represented in terms of a

rotation scheme and a signature In this section we examine embeddings in

which a p ortion of the rotation has b een preordained We introduce this

topic with nonorientable embeddings

Supp ose that we are given a graph G and edges a uv b uw What

can we say ab out embeddings in which a is adjacent to b in the lo cal rotation

at u To answer this question create a related graph G by adding in a new

edge v w which lies alongside a and b so that v uw is a face of the embedding

Any embedding of G where v uw is a face corresp onds to an embedding of G

with a adjacent to b in the lo cal rotation Thus the lo cal rotation constraint

on G is equivalent to the xedface constraint on G

If we are given a set of restrictions on the lo cal rotations then we can

iterate the ab ove pro cess to form a related graph enco ding all of the restric

tions If an edge is unaected by the restrictions then we can replace it by a

digon which forms one of the distinguished walks Thus we can assume that

every edge is in exactly one of the distinguished walks

This motivates the following denition A relative graph is a graph G

together with a collection of walks which partition the edge set A relative

embedding of G is an embedding of the underlying graph such that the

distinguished walks are face b oundaries Each edge lies on two faces exactly

one a distinguished walk A relative graph G is a fat graph if each of the

xed faces are of length two In this case we can recover the underlying graph

G by replacing each digon with a single edge Relative embeddings of G are

equivalent to the usual embeddings of G

In the orientable case we require the underlying relative graph to have

a xed direction on each edge The distinguished walks must resp ect these

directions The embeddings are those in which the directed walks are faces

in the oriented surface

Relative embeddings have proven useful in studying the amalgamations

of graphs the distribution of embeddings and in re

embedding theorems Bonnington has shown the relative analogue

of Xuongs theorem and Archdeacon Bonnington and Siran have shown

the relative version of Neb eskys Theorem

Signed Embeddings

Recall that an embedding in a nonorientable surface is characterized by a

signature and a rotation What can b e said ab out embeddings if the signature

is preordained Dene a signature on a closed walk as the pro duct of the

signatures on its edges In an embedding using this signature a walk is

orientation reversing if and only if it is signed minus So an embedding of

a signed graph can b e interpreted as preordaining the orientation preserving

orientation reversing walks

For the general theory of signed embeddings we refer the reader to the

works of Zaslavsky We mention three particular results of

interest to the author Zaslavsky has determined the maximum Euler

genus among all signed graphs on n vertices where every edge is signed neg

atively This is given by dnn e where n He has also

characterized by forbidden minors the pro jective planar signed graphs Siran

has investigated the sp ectrum of the signed genus and notes that no

interpolation theorem like Dukes holds in this setting

Embeddings in space

To date our fo cus has b een to depict graphs in top ological spaces which are

for the most part lo cally dimensional This is due in part to the following

theorem

Theorem Any simplicial graph embeds in space with al l edges straight

lines

We do not require arbitrary embeddings to b e rectilinear but to avoid

pathologies we do require embeddings to b e piecewise linear

The ab ove theorem shows existence but there is much that can b e said

ab out embeddings in space One question is when can the graph b e em

b edded so that the cycles are all nice There are various dierent notions

of nice we give three An embedding is linkless if no two disjoint cycles

are linked in space that is if they can b e pulled apart An embedding is

knotless if each cycle is unknoted that is if the fundamental group of the

complement is free An embedding is at if each cycle b ounds a disk disjoint

from the rest of the graph Note that at implies linkless and knotless

When do es a graph have a linkless knotless or at embedding Sachs

and Conway and Gordon showed the following

Theorem Any embedding of K in space contains a pair of disjoint

cycles which are linked

Rob ertson Seymour and Thomas proved that a graph admits

a at embedding if and only if it admits a linkless embedding settling a

conjecture of Bohme So while the two concepts are dierent for sp ecic

embeddings the class of graphs so embeddable are equivalent More strongly

they developed a theory of spatial embeddings in which they proved that if

two at embeddings are not ambient isotopic then they dier on a sub di

vision of K or of K They also showed that any at embedding can b e

5 33

transformed to any other at embedding by a series of switches similar

to Whitney switches for planar graphs Finally they showed that a graph

admits a at embedding if and only if it do es not contain as a minor one

of the seven graphs These seven graphs are the Y equivalents of the

Petersen graph

Negami has given a type of Ramsey theorem for knots

Theorem Let N be a knot There exists an f N such that any drawing

of K with straight line segments contains a cycle which is equivalent to

f (N )

Notice here that rectilinear embeddings are necessary Otherwise each

edge could b e embedded in a very knotty fashion so that every cycle was a

very complicated knot type A similar theorem for complete bipartite graphs

is given by Miyauchi

Pseudosurfaces

A pseudosurface is a top ological space formed from a not necessarily con

nected manifold by taking a nite number of p oints partitioning these

p oints into parts then top ologically identifying the p oints within each part

The p oints which are not lo cally R are called pinch points For example

the spind le surface is formed by identifying the north and south p oles on a

sphere Equivalently it is formed from the torus by identifying a noncon

tractible cycle to a p oint The banana surface is formed from two spheres

by identifying their north p oles to a single p oint and their south p oles to a

second p oint The surface lo oks like two bananas joined at their b ottoms as

well as at their tops

Embeddings of graphs into pseudosurfaces usually carry the restriction

that the pinch p oints are vertices of the embedded graph They can b e

describ ed combinatorially using lo cal rotations that are simply p ermutations

of the incident edge ends not necessarily cyclic p ermutations The number

of cycles in the p ermutation is the number of disks meeting at that p oint

Many of the questions asked of surfaces have also b een asked of pseudosur

faces For example do es there exist a nite set of graphs whose top ological

exclusion characterizes embedding of a particular pseudosurface Bo dendiek

and Wagner have shown that the answer is yes for the spindle surface

Siran et al showed that the answer is no for the banana surface

nding an innite class of graphs which do not embed on the banana sur

face but such that each prop er subgraph do es so embed however there are

only irreducible graphs of connectivity In this context note that em

b edding on a pseudosurface is not hereditary under minors b ecause of the

restriction that vertices must lie on pinch p oints An edge joining two pinch

p oints cannot b e contracted in the pseudosurface Hence Sirans result do es

not contradict the Rob ertson and Seymour pro of of Wagners conjecture

Coloring graphs embedded on psuedosurfaces or families of pseudosur

faces has a rich history We rst examine pseudosurfaces formed from the

sphere by identifying p oints pinched spheres An M pire is a graph embed

ded on a pinched sphere where each pinch p oint corresp onds to at most M

spherical p oints What is the maximum chromatic number of all M pires

The name arises from the dual map coloring problem where each country

empire may have as many as M comp onents all of which must receive the

same color

Heawoo d showed an upp er b ound not surprisingly based on Euler

Characteristic of M for M and gave a map with regions broken

into M pires for M The upp er b ound is exact as shown by the

constructions of Jackson and Ringel Note that the b ound is wrong for

M by the colortheorem

We next examine graphs formed from two spheres by identifying pairwise

p oints in the rst with p oints in the second We express the appropriate

coloring problem in dual form Consider two planar graphs G on the

earth and G on the mo on together with a bijection F G

M E

F G An earthmoon coloring assigns colors to the faces of b oth graphs

such that in b oth graphs adjacent faces receive distinct colors but f and

f receive the same color The idea is that each country on the earth has

a colony on the mo on and a country and its colony should receive the same

color What is the maximum chromatic number of all earthmo on graphs

The problem is equivalent to the dual problem of determining the maximum

chromatic number of all graphs of thickness two An Euler argument shows

that colors suce an example due to Sulanke cf needs colors The

b ound has not b een tightened further

We refer the reader to for exp ository articles on earthmo on

colorings and M pires

Lo cally Planar Embeddings

The class of planar graphs is one of the most imp ortant in graph theory

Some graphs are not planar but they are if you lo ok closely enough For

example consider C C embedded in the torus in a natural manner

100 100

The graph is not planar however if you x a vertex and lo ok at a lo cal

neighborho o d the embedding lo oks planar In this example the subgraph

induced by all vertices of distance at most from a xed vertex is planar

The lo cal planarity of the surface is reected by the fact that such large lo cal

neighborho o ds of vertices are planar It is hop ed that such lo cally planar

graphs share prop erties with planar ones

We develop three dierent measures of the lo cal planarity of a graph G

embedded in a surface S The edgewidth of the embedding ew G is the

length of the shortest walk in the graph which is nonhomotopically null in the

surface Such a walk is necessarily a simple cycle In fact it is the shortest

simple cycle which do es not b ound a disk in the surface This parameter

was rst introduced by Thomassen The dualwidth dw G is the

edgewidth of the dual embedding The facewidth f w G is the minimum

n C G taken over all noncontractible C in the surface A cycle C achieving

this minimum can b e chosen to b e simple and intersecting only vertices of

the graph The facewidth of the graph is also known as the representativity

of the embedding The idea b ehind representativity is that the parameter

measures how well the embedded graph represents the surface where the idea

b ehind the width is that the embedded graph measures how wide the handles

are in terms of the graph A third p oint of view is that these parameters

measure the density of the graph embedded on the surface Finally this

parameter measures the lo cal planarity of the embedding Observe that the

facewidth of an embedding is equal to the facewidth of the dual

The facewidth was rst introduced by Rob ertson and Seymour The

rst extensive study of this parameter was by Rob ertson and Vitray

Several sp ecial cases have b een commonly used For example an embedding

of a connected G is of f w if and only if it is cellular Likewise embeddings

of connected graphs with f w have b een called circular each face is

b ounded by a simple cycle or closed cell CTC the closure of each face is

a closed cell Embeddings of connected graphs with f w are called

polyhedral In a p olyhedral embedding the face b oundaries are all induced

nonseparating cycles

In the following subsections we discuss embedded graphs which are min

imal with resp ect to these width parameters what these parameters tell us

ab out other embeddings coloring lo cally planar graphs and how to nd

cycles of a sp ecial homotopy type

Minimal Embeddings

Let G b e an embedding of a graph with facewidth k This embedding is

f w k minimal if every embedded minor of G has f w k A f w k

minimal embedding is connected and has the prop erty that the deletion or

contraction of any edge will lower the facewidth It follows from Rob ertson

and Seymours pro of of Wagners Conjecture discussed in Section that

for each xed surface S there are only nitely many f w k minimal

embeddings Malnicand Mohar proved this directly when k The

general case was also shown directly by Malnicand Nedela and by Gao

Richter and Seymour

Barnette and indep endently Vitray found the k minimal graphs

for the pro jective plane Barnette has also discussed various

ways to generate triangulations p olyhedral and closed cell maps in simple

surfaces

Randby cf has shown that any f w k minimal embeddings in

the pro jective plane can b e obtained from a certain k k pro jective grid by a

sequence of Y transformations Schrijver has shown that for the torus

there are exactly k k k o dd classes of k minimal embeddings up

to Y and Y transformations The corresp onding number for k even is

k k

Hutchinson has shown that any triangulation with facewidth

k of the surface with genus g has at least ck g log g vertices Przytycka

and Przytycki gave such constructions with ck g log g vertices the

constants are dierent

Schrijver examined the following renement of facewidth A kernel is

an embedded graph such that any edge deletion or contraction decreases the

facewidth in some homotopy class that is the global facewidth may not b e

decreased but for some curve C the minimum G C over all C homotopic to

C decreases The name comes from doing deletions and contractions which

change the width of no homotopic class until arriving at a core minor where

no further such op erations are p ossible Schrijver observed that taking

the dual and doing Y and Y transformations do not change the width in

any homotopy classes Conversely he showed that any two embedded

kernels with the same width in each homotopy class were equivalent up to

duality Y and Y transformations

Reembedding Theorems

A connected graph has at most one planar embedding Are embeddings

which are lo cally planar necessarily unique Are they necessarily minimum

genus

In one sense the answer to the two questions ab ove is yes Rob ertson and

Vitray proved that a connected graph embedded in a surface of genus

g with facewidth greater than g is necessarily a unique minimal genus

embedding Seymour and Thomas p ersonal communication have improved

this to O log g and then to log g log log g Mohar has similar

results

In another sense the answer to the two questions ab ove is no The b ounds

ab ove dep end on the surface Rob ertson and Vitray conjectured that a

constant b ound suced for all surfaces in particular However Thomassen

and indep endently Barnette and Riskin found simple counterexam

ples involving toroidal graphs with nontoroidal embeddings of facewidth

Eg in a toroidal C C take n nonadjacent faces and n homotopic dis

4 n

joint essential cycles as the n face b oundaries in a facewidth nontoroidal

embedding Rob ertson and Vitray then raised the conjectured b ound to

However Archdeacon constructed nconnected graphs G with

two dierent embeddings of facewidth n The surfaces involved can either

b e the same violating uniqueness or dierent violating genus

Rob ertson and Vitray and indep endently Thomassen observed

that any nonplanar embedding of a planar graph has facewidth at most

two Mohar Rob ertson and Vitray characterized embeddings of planar

graphs in the pro jective plane This was extend to other surfaces in

Mohar showed that apex graphs graphs G with G v planar for some

v have no nonorientable facewidth three embeddings but they do have

orientable ones

Various authors have examined the

uniqueness of embeddings on the pro jective plane Klein b ottle and torus

In particular we note that Lawrenchenko and Negami have found all

graphs which triangulate b oth the torus and the Klein b ottle

The following amazing theorem of Fiedler Huneke Richter and Rob ert

son gives the orientable genus of all pro jective planar graphs

Theorem Let G be a projective planar graph embedded with facewidth

n Then the orientable genus of G is bnc

In particular note that any two embeddings of a pro jective planar graph

must have facewidth that diers by at most one Rob ertson and Thomas

have a similar result giving the orientable genus of graphs on the Klein

b ottle

Thomassen has examined largeedgewidth or LEWembeddings

These are ones in which the edgewidth exceeds the length of the longest

face He has shown that these embeddings are genus embeddings are unique

if the graph is connected have only Whitneytype switches if the graph is

connected and gives a p olynomialtime algorithm for nding such embed

dings if they exist

Coloring Lo cally Planar Graphs

Planar graphs can b e colored Can lo cally planar graphs b e colored No

Fisk constructed graphs of arbitrarily large facewidth with chromatic

number ve But four is close as shown by Thomassen

Theorem A graph embedded on the orientable surface of genus g with

14g +6

edgewidth at least is colorable

The pro of involves cutting along a set of cycles to reduce to a planar

graph then invoking a sp ecial version of the color theorem In a similar

manner Hutchinson has shown that a graph embedded with large face

width and all faces of even length is colorable

Finding Cycles in Embedded Graphs

One use of large facewidth embeddings is to guarantee cycles of a certain

homotopy type We examine some results of this type

When do es an embedded graph contain a noncontractible separating cy

cle Clearly the surface involved must b e of genus at least two Barnette

conjectured that any simple triangulation has such a cycle Indep endently

and more generally Zha conjectured that facewidth suces The b est

known result is due to Zha and Zhao who showed that facewidth

suces see also Richter and Vitray and Brunet Mohar and Richter

Brunet Mohar and Richter have shown that an embedded graph with

face width w contains at least bw c disjoint noncontractible homotopic

cycles Schrijver improved this to bw c for the torus

Thomassen showed that any graph embedded on the surface of genus

g with facewidth at least has a set of g disjoint cycles which can

b e cut along to form a planar graph Graaf and Schrijver proved that

every facewidth w toroidal graph contained a C C minor

b2w 3c b2w 3c

We mention the following on cycles in graphs unrelated to homotopy

Whitney proved that every connected plane triangulation is Hamiltonian

A number of authors have investigated

generalizations of this concept relaxing the connectivity replacing planarity

with lo cally planar on other surfaces and replacing Hamiltonian with various

types of walks We refer the reader to the survey by Ellingham in this volume

Graph Minors

Recall that a graph H formed from G by a sequence of deleting isolated ver

tices and deleting or contracting edges is called a minor The theory of graph

minors has recently played a central role in top ological graph theory The

prominence is due in most part to the recent pro of of Wagners Conjecture

by Rob ertson and Seymour

Theorem In any innite set of graphs one is a minor of another

The theoretical and algorithmic implications of this result are enormous

We illustrate this by rephrasing the Rob ertsonSeymour theorem in terms of

graph prop erties

A prop erty is hereditary with resp ect to an order if whenever G has

the prop erty then any H G has that prop erty For example the prop erty

of embedding on a xed surface is hereditary under the minor or top ological

orderings In an order without innite descending chains any graph G not

p ossessing a hereditary prop erty contains an H G which is minimal with

out that prop erty ie H do es not have the prop erty but every K H do es

have the prop erty For example we can talk ab out graphs that are minor

minimal with resp ect to not embedding on a xed surface The set of these

minimal elements are pairwise noncomparable Hence Rob ertsonSeymours

Theorem implies the following

Theorem For any hereditary property P there is a set M P such that

G has property P if and only if it has no minor in M P

As a corollary we obtain the p owerful generalization of Kuratowskis The

orem mention in Section

Corollary For each surface S a graph G embeds in S if and only if it

has no subgraph homeomorphic to one in a nite collection M S

The Rob ertsonSeymour theorem is sometimes stated in terms of orders

A partial order is a wel lq uasiorder if for any sequence G G there

1 2

is an i j such that G G A wellquasiorder has no innite strictly

i j

decreasing sequence If a partial order has no strictly decreasing sequence

then b eing wellquasiordered is equivalent to having no innite set of non

comparable elements Since graphs under the minor order have no innite

decreasing chains the Rob ertsonSeymour Theorem asserts that this is a

wellquasiordering

Trees and treelike graphs play an imp ortant role in the theory of minors

The algorithmic implications of the Rob ertsonSeymour Theorem are very

imp ortant We examine these two asp ects in the following subsections

Figure A graph of treewidth three

Trees and TreeWidth

Kruskal proved in that ro oted nite trees were wellquasiordered

under the top ological containment order It follows that there are also no

innite antichains in this collection under the minor order Wagner

p ointed out that the collection of all graphs do es contain an innite antichain

b e the graph obtained in the top ological order Namely for each n let C

is from the n cycle by replacing each edge with two edges in parallel No C

top ologically contained in another so these are an antichain This collection

2 2

is a minor of C do es not violate the Rob ertsonSeymour Theorem since C

n m

whenever m n

If trees are wellquasiordered maybe so are treelike graphs We use the

following measure of how closely a graph resembles a tree A graph G is a

K cockade if there is a sequence of subgraphs G G such that each G

n 1 m j

is a complete graph on n vertices and G G G is contained in

j 1 j 1

some G for i j In other words a K co ckade is formed by rep eatedly

i n

adding in copies of a xed complete graph on n vertices by identifying some

th th

of the vertices in the j copy with those in an earlier i copy The treewidth

of a graph H tw H is the smallest n such that H is a subgraph of some

K co ckade Note that a graph is a tree if and only if it is of treewidth

n+1

If H is a minor of G then tw H tw G

The ngrid is the planar graph P P For every planar graph G there

n n

is an n nG such that G is a minor of the ngrid The assumption of

planarity is necessary since every minor of a planar graph is planar The

pro of of this result is easy conceptually imagine the graph drawn in the

plane with disks for the vertices Place a very ne grid on the pap er Deform

the line segments until they follow horizontal and vertical lines on the grid

A sub division of the graph is now a minor of the grid after contracting all

edges within the vertex disks

The ngrid is related to the width of the graph The ngrid has large

treewidth Hence any graph which contains the ngrid as a minor also

has large treewidth The TreeWidth Theorem asserts the converse

Theorem There exists a function f such that a graph G has

tw G f k if and only if it has a k grid minor

As a partial result towards Wagners minor conjecture Rob ertson and

Seymour were able to show that the class of graphs of b ounded tree

width were wellquasiordered under minors Using that result and a struc

ture theorem they were able to prove the general case

Algorithmic Implications

The graph minors pro ject has pro duced some fascinating results on structural

prop erties of graphs and pro ofs of some farreaching fundamental theoreti

cal results At the same time it has imp ortant algorithmic implications in

theoretical computer science We briey discuss these implications in this

section

A fundamental quest in computer science is to nd ecient algorithms

for problems Many problems are thought to b e hard in the sense that there

are no algorithms to solve the problem which run in a time b ounded by a

p olynomial in the size of the input In fact the existence of p olynomialtime

algorithms for a large class of problems is known to b e equivalentthese

problems are called NPcomplete Since the general b elief not proven is

that these algorithms are hard in general it is interesting to nd classes of

graphs which do have such p olynomialtime algorithms

A number of problems which are NPcomplete in general are p olynomial

for graphs of b ounded treewidth Here the algorithm is able to exploit

the treelike structure of the graph For example Arnborg has found

p olynomial time algorithms for k coloring and Hamiltonicity of graphs of

b ounded tree width

A fundamental result due to Rob ertson and Seymour is the following

solution to the k path problem

Theorem For each xed k there exists a polynomial time algorithm for

deciding if a graph G with vertices x y x y has k disjoint paths P

1 1 k k i

each joining x to y

i i

As a corollary to the ab ove is the existence of a p olynomialtime algorithm

for testing if a xed graph H is a minor of an input graph G A mo dication

of the algorithm also tells if H is a top ological subgraph of G in p olynomial

time

A combination of the k path algorithm and the solution of Wagners con

jecture leads to a very p owerful result There are a nite number of minor

minimal graphs whose exclusion determines when a graph has a hereditary

prop erty Testing for each of these H as a minor of G can b e done in p oly

nomial time Hence we get the following result

Theorem For any property hereditary under the minor order there is a

polynomialtime algorithm to test if a graph has this property

This implies among other things a xed degree p olynomialtime algo

rithm for testing embeddability into a xed surface and a p olynomialtime

algorithm for testing if the treewidth of a graph is less than a xed constant

Random Topological Graph Theory

To date much attention has b een fo cused on the minimum and maximum

genus of a graph But what do es a typical embedding lo ok like Because we

describ e our embeddings combinatorially in terms of rotations and signatures

it is p ossible to pick an embedding at random What should we exp ect the

drawing to lo ok like

White has describ ed ve mo dels for random top ological graph the

ory In the rst mo del you x the graph G and select a rotation uniformly

at random here the embeddings are all orientable One goal is to study the

distribution over all embeddings of the genus of the surface This embedding

distribution was rst introduced by Gross and Furst The complete

embedding distribution is known only for a few small graphs and for a few

innite classes The latter include b ouquets see also closed

ended ladders and cobblestone paths

It is interesting to note that all known embedding distributions are uni

mo dal in fact strongly unimo dal It is conjectured that this is always the

case

Short of calculating the entire embedding distribution the next parameter

of interest would b e the exp ected value of the genus also known as the

average genus Stahl has given upp er b ounds on the exp ected number

of regions These translate to lower b ounds on the average genus Using

this Lee has shown that if the number of edges is asymptotic to cn

where n is the number of vertices then the average genus is asymptotic to

the maximum In particular this holds for many classes such as complete

and complete bipartite graphs Stahl do es some calculations which

show that the average genus is roughly linear for linear families of graphs

These are graphs made up of a chain of comp onents arranged in a pathlike

manner where each comp onent shares only a few vertices with its neighbors

Both Lee and Stahls results indicate that the average genus is linear in the

number of edges This was shown for simplicial graphs by Chen and Gross

Their work also shows that the set of values of the average genus for

connected or for connected simplicial graphs has no limit p oints This

work leads to a lineartime algorithm for testing isomorphism of graphs from

a class with b ounded average genus

A second mo del includes nonorientable embeddings Here we x the graph

and select a rotation and a signature at random White shows that in

(G)

this mo del the probability that an embedding is orientable is In other

words for most families of graphs including complete and complete bipartite

graphs almost all embeddings are nonorientable

A dierent approach would b e to consider the sample space of all lab elled

graphs on n vertices with edges o ccurring indep endently with probability

pn The rst goal here is to nd the exp ected value of the minimum genus

of the graph This was rst done by Archdeacon and Grable who showed

that for suciently large pn including constant probabilities the minimum

genus was roughly pn the b ound given by Eulers formula for triangular

embeddings This was also proven by Rodl and Thomas who rened

the b ounds on the edgeprobability

We close by noting that Bender Gao and Richmond have shown

that for each xed surface almost all ro oted embeddings of graphs with m

edges have facewidth O log m

Symmetrical Maps

In the study of mathematics highly symmetric ob jects are widely studied

Indeed their symmetries are considered b eautiful The same is true in top o

logical graph theory Let G b e a connected graph embedded on an oriented

surface or an oriented map for short An automorphism of the map G is

a function from vertices to vertices and edges to edges that preserves inci

dence and resp ects the rotation That is it is an automorphism of the graph

which extends to an orientationpreserving automorphism of the surface

Equivalently it is a function which preserves oriented region b oundaries

How many automorphisms can an oriented map have We claim that it

can have as many as E In particular x a directed edge x y Supp ose

that arc is carried to the edge u v This target u v determines the whole

automorphism The idea b ehind the pro of is that preserving the lo cal ro

tation determines where each edge adjacent to x y maps and this is then

extended to the whole graph by connectivity A oriented map is called regu

lar if the order of its automorphism group is E since acting on arcs the

group has order equal to the degree

A reection of an oriented map is an isomorphism b etween that oriented

map and its mirror image the embedded graph with the opp osite orienta

tion on the surface The extended automorphism group allows reections as

well as orientationpreserving automorphisms It is known that the automor

phism group of the map is a subgroup of index one or two in the extended

automorphism group It follows that if a regular map admits a reection

then its extended automorphism group is of order E Maps achieving this

b ound are called reexible

In the nonorientable case a map is regular if and only if its automorphism

group is of order E There are two automorphisms xing a directed edge

the identity and a second swapping the faces on either side of that edge

There is some confusion in terminology in the literature regarding when a

map is regular Some authors esp ecially those working with b oth orientable

and nonorientable surfaces prefer to call an orientable map regular only if

the extended automorphism group is as large as p ossible reserving the words

rotary or orientably regular for what Ive called here regular nonreexible

maps The terminology here is the same as that of Coxeter and Moser

and Jones and Singerman

There are three main ways to approach regular maps x the graph

involved x the automorphism group involved or x the surface in

volved

Construction techniques for the rst two frequently involve Cayley graphs

In a Cayley graph the vertex set is the set of elements in some group and

the edges are g g h for g and h for some set This

is a generating set if it generates the group or equivalently if the resulting

Cayley graph is connected In a Cayley graph the group acts transitively on

the vertices An embedding of a Cayley graph commonly uses a xed cyclic

p ermutation on to dene the lo cal rotation Dep ending on prop erties of

the group generating set and the cyclic rotation the embedding may b e

regular

We do not examine Cayley graphs further in this section but instead

refer the reader to Whites survey in this volume We do however mention

that James and Jones have completely classied the orientable maps

based on complete graphs We also mention that several pap ers

have examined lifting automorphisms from an embedded voltage graph to

its derived covering map leading to some nice constructions This technique

involves some but not all of the algebraic p ower of Cayley graphs

We turn our attention to regular maps on a xed surface Hurwitz exam

ined nite sets of homeomorphisms of a surface with itself He showed the

following

Theorem Every nite homeomorphism group of S g with itself

has order at most g

Tucker related the general problem of nite homeomorphism groups

to graphs with the following theorem

Theorem For any nite group H of homeomorphisms of S there ex

ists a Cayley graph G on the vertex set H embedded on S such that each

isomorphism of G extends to a homeomorphism of S

The preceding two theorems do not imply that there are only nitely

many Cayley graphs of each genus g Instead Wormald see

constructs innitely many Cayley graphs of genus despite the fact that

this surface has only nitely many homeomorphisms groups

Babai and indep endently Thomassen have shown that this

b ehavior is an anomaly of S In particular they showed that there are only

nitely many vertextransitive graphs of given genus g

Yet another kind of symmetry for an embedded graph is selfduality that

is the embedded map is isomorphic to the dual map A slightly weaker

version is to require that G and G b e isomorphic as graphs but allows

their embeddings to b e dierent Some interesting results on selfdual pla

nar graphs are given by Archdeacon and Richter and by Servatius and

Servatius We refer the reader to the survey by Archdeacon

Combining these types of symmetry Archdeacon Siran and Skoviera

have constructed classes of selfdual regular maps Payley maps cf

have the remarkable prop erty of b eing regular selfdual and self

complementary They also give rise to selfdual partially balanced incomplete

blo ck designs

Ten Problems

Despite the plethora of results presented there remain many interesting ques

tions We oer the following unsolved problems

Problem Find an easy ie noncomputer proof of the ColorTheorem

Problem Show that every edgeconnected graph has a set of simple

cycles which together contain every edge exactly twice

Problem Show that every edgeconnected cubic graph has a set of

six perfect matchings which together cover every edge exactly twice

Problem Show that every simple triangulation of an orientable surface

can be edgecolored so that each color appears on each face

Problem Find the earthmoon coloring number

Problem Find the crossing number of K

Problem Find the genus of K

pq r

Problem Show that a graph has a planar cover if and only if it embeds

in the projective plane

Problem Find a regular map for each nonorientable surface

Problem is called the Cycle Double Cover Conjecture It was inde

p endently p osed by Seymour and by Tutte It is equivalent to claiming that

every graph has an embedding such that the dual is lo opless Problem

Fulkersons Conjecture is a type of dual to Problem

David Craft has found many of the embeddings needed for Problem

Note that if p q r then the number of triangles is at most q r Assuming

all other faces are quadrilaterals gives the conjectured b ound through Eulers

formula

Regarding Problem a graph G covers H if there is a graph map

from G to H which is an isomorphism on the neighborho o d of each vertex

A graph embedding in the pro jective plane has a fold planar cover Since

having a planar cover is preserved under minors it suces to show that the

minorminimal nonpro jectiveplanar graphs do not have planar covers

There are two remaining cases K K and K K whose pro of would

7 2 44 2

complete the general result

Conclusion

I must express my regret at the vast areas of top ological theory which I was

not able to cover in this survey Several areas in particular are worthy of

inclusion in any survey

I direct the reader to Carsten Thomassens wonderful work using graph

theory to prove results in top ology Among his results are graph theoretic

pro ofs of the Jordon Curve Theorem a deep er understanding of the

relationship b etween the Jordon Curve Theorem and Kuratowskis Theorem

and a nice pro of that every surface admits a triangulation

Another area not included is the dual theories of current and voltage

graphs This techniques uses quotient structures under group actions to

describ e embeddings This allows an economical description of embeddings

It proved essential to the pro of of the Map Color Theorem The reader is

referred to the seminal articles by Gross and Alp ert see also

A more leisurely introduction is the b o ok by Gross and Tucker

Other surveys on top ological graph theory are presented by White and

Beineke and Carsten Thomassen The reader is referred to

for additional results on snarks Joan Hutchinson gives a nice

exp osition on map colorings empires and the earthmo on problem In

Mohar gives a survey of results on lo cal planarity Rob ertson Seymour and

Thomas survey linkless embeddings in

The b o ok by Jensen and Toft on graph colorings cannot b e recom

mended highly enough

I learned a lot writing this pap er I hop e that the reader has also found

it informative

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