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 2 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 m 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 5 ding Using Euler genus for the sphere and girth g V and E for K we violate the preceding inequality This contradiction 5 shows no such planar embedding exists A similar argument works for K 33 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 M 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