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Graphs on Surfaces

- We are mainly interested in embeddings of graphs on surfaces:
- h: G ! S.
- An embedding should be differentiated from immersion.
- On the left we see some forbidden cases for embeddings.

Cellular (or 2-cell) embedding

- Embedding h:G ! S is cellular (or 2-cell), if S \ h(G) is a union of open disks.
- A 2-cell embedding is strong (or proper) if the closure of each open disk is a closed disk.
- Proposition: Only connected graphs admit 2-cell embeddings..
- On the left we see two embeddings of K4 in torus S1. The first one is cellular, the second ons is not!

2-Cell Embeddings and Maps

- 2-cell embeddings of graphs are also known as maps. There is a subtile difference in the point of view.
- In the former the emphasis is given to the graph while in the latter the emphasis is in the map, a structure, composed of vertices, edges and faces. Examples of maps include surfaces of polyhedra.
- Maps include different, equivalent, cryptomorphic purely combinatorial definitions that can be used as a foundation of a theory of maps that is independent of topology.

Genus of a Graph

- Let g(G) denote the genus of a graph G. This parameter denotes the minimal integer k, such that G admits an embedding into an orientable surface of genus k.
- Note: (G) = 0 if and only if G is planar.

Euler Characteristics

- To each closed surface S we associate a number (S) called Euler characteristics of S.
- (Sg) = 2 – 2g, for orientable surface of genus g.
- (Nk) = 2 – k, for non-orientable surface of crosscap number (non-orientable genus) k.

Euler Formula

- Let G be a graph with v vertices, e edges cellularly embedded in surface S with f faces. Then
- v – e + f = (S).

Rotation Scheme

- Let G be a connected graph with the vertex set V, with arcs S and edges E. For each v 2 V define the set: S[v] = {s 2 S| i(s) = v}. Let and be mappings:
- r: S ! S
- l: S !{-1,+1}.
- with the property:
- Permutation r acts cyclically on S[v], for each v 2 V.
- l(s) = l-1(r(s)), for each s 2 S. [Hence is a voltage assignment. In our case: l(s) = l(r(s))].
- The triple (G,r,l) is a called a rotation scheme, defining a 2-cell embedding of G into some surface.

Interpretation of Rotation Scheme

- We follow arcs starting at s0 until we return to the initial arc.
- s Ã s0,
- s Ã(s).
- positive Ã True.
- While s s0 do
- If positive then
- If (s) = 1 then
- s Ã(s)
- else
- positive Ã False;
- s Ã(s)-1
- else
- If (s) = 1 then
- s Ã(s)-1
- else
- positive Ã False;
- s Ã(s)

r(s)

r2(s)

s

r(s)

r3(s)

r4(s)

r(r(s))

r(s)

r2(s)

s

r(s)

r3(s)

r4(s)

r(r(s))

Rotation Scheme and Rotation Projection

- Rotation scheme can be represented by rotation projection.
- Rotation r can be reconstructed from the bottom drawing. Each arc s carries l(s) = 1.

Example

- On the left we see the rotation projection of K4. The faces are triangles.
- There is no cycle with an odd nunber of “crosses”.
- V – E + F = 4 – 6 + 4 = 2.
- The surface is a sphere!
- Exercise: Analyse the faces of the embedding if all crosses are removed from the figure on the left.

Main Fact

- Theorem: Any 2-cell embedding of a graph G into a surface S can be described by a rotation scheme (G,,). Furthermore, by face tracing algorithm the number of faces F can be computed yielding (S). Finally, S is non-orientable if and only if G contains a cycle
- C = (e1,e2, ... , ek) such that
- (C) := (e1) (e2) ... (ek) = -1

Combinatorial Theory of Maps

- There are several cryptomorphic definitions of maps (graphs on surfaces.)
- Rotation schemes represent such a tool.
- Note that we start with a graph G and add additional information (G,,) in order to describe its 2-cell embedding. In some closed surface.
- We may also start directly from maps or polyhedra.

v

e

Flag Systems- Let V,E,F be disjoint (finite) sets.
- Fµ V £ E £ F is a flag system. Here:
- V vertex set,
- E edge set
- F face set.
- A face that is a polygon with d sides, (a d-gon), consists of 2d flags (see figure on the left!)

Flag Systems are General

- Using flag systems we can describe general complexes such as books.
- Note the a 3-book contains a non-orientable Möbious strip.

Flag systems from 2-cell embeddings

- To a 2-cell embedding we associate a flag system as follows. Let V be the set of vertices, E, the set of edges and F the set of faces of the embedding. Define
- µ V £ E £ F as follows:
- (v,e,f) 2 if and only if v, e, and f are pairwise incident.

The 1-skeleton of a flag system.

- Given a flag system µ V £ E £ F, we may study its projection to the first two factors:
- A = {(v,e)| (v,e,f) 2}.
- Define:
- i:A ! V by i: (v,e) v and
- Ve = {v 2 V| (v,e) 2 A}.
- Assume |Ve| · 2, for each e 2 E.
- We may define r:A ! A by:
- r(v,e) = (w,e) if Ve = {v,w} and
- r(v,e) = (v,e) if Ve = {v}.
- The quadruple (V,A,i,r) is a pre-graph. It is called the 1-skeleton of .
- Given there is an easy test whether the 1-skeleton is indeed a graph: for each e 2 E we must indeed have |Ve| = 2.

1-co-skeleton

- If we replace the role of V and F in a flag system µ V £ E £ F we obtain a 1-co-skeleton.
- We say that the skeleton and co-skeleton are dual graphs.

Homework

H1: If one of 1-skeleton is a graph is the 1-co-skeleton a graph too? Prove or find a counterexample.

Exercises

- N1. Determine the flag system describing the four-sided pyramid.
- N2. Determine the 1-skeleton and 1-co-skeleton for N1.
- N3. Define the notion of automorphism of a flag system . For the case N1 find the orbits of Aut .

When does a flag system define a surface?

- As we have seen in the case of a book we may have an edge belonging to more than two faces. This clearly violates the rule that each point on a surface has a neighborhood homeomorphic to an open disk.
- Therefore a necessary condition is:
- Each for each flag (v,e,f) 2 there must exist a unique triple (v’,e’,f’) 2 V £ E £ F with v’ v, e’ e, f’ f such that (v’,e,f), (v,e’,f),(v,e,f’) 2.
- Another obvious condition is that the 1-skeleton must be connected.
- However, a flag system satisfying these two conditions may still represent more general spaces than surfaces.
- It may represent a pseudosurface.
- Let us define:
- v = {(f,e)| (v,e,f) 2}.
- e = {(v,f)|(v,e,f) 2}.
- f = {(v,e}| (v,e,f) 2}.
- Each of the three structures defined above can be represented as graph. More presicely, each of them is regular 2-valent graph.
- is a surface if and only if each graph v, e and f is connected.

Limits of flag systems

- Unfortunately, there are connected graphs whose 2-cell embeddings cannot be represnted by flag systems.
- Proposition. Let G be a connected graph. If G contains a loop or a bridge no 2-cell embedding of G can be described by flag systems.
- [A bridge is an edge whose removal disconnects the graph.]

Some limits of flag systems

- On the left we see K4 embedded in torus with one 4-gon and one 8-con.
- Green and red flag have all three matching components equal.
- This map cannot be described by flag systems.

Self-avoiding maps

- Theorem: A 2-cell embedding of G in some surface can be described by a flag system if an only if neither G nor its dual contains a loop.
- A map that satisifies the conditions of this theorem will be called self-avoiding.

Flags, from a different view-point.

- Let us forget about V,E, F for a moment. Let the set of flags F be given.
- For instance, on the left, we see them as triangles.
- Define the flag graph G(F):
- V() = .
- f ~ f’ if and only if triangles have a common side.

From flags to flag graph.

First the vertices.

From flags to flag graph.

First the vertices.

Next: three kinds of new edges:

along the edges

across the edges.

across the angles.

Flag graphs for 2-cell embedded graphs.

Flag graph is:

- connected

- trivalent

- contains a 2-factor of form m C4.

Flag graphs for 2-cell embedded graphs.

A practical guide to the construction.

The first step when rectangles are placed on each edge is shown.

Yet another view to flag graphs.

- We may start with three involutions:
- t0, t1, t2 : F!F
- 02 = 12 = 22 = 1, each fixed-poit free.
- t0t2 = t2t0, also fixed-point free.
- Each invoultion corresponds to a 1-factor. Together they define a cubic graph: the flag graph ().
- The group <0,1,2>, called monodromy group must act transitively on . [This is eaquivalent to saying that () is connected.]
- These axioms define a (combinatorial) map on a surface.

Combinatorial Map.

- Combinatorial map is defined by three involutions satisfying the axioms from the previous slide.
- Orbits of <2,1> acting on define V.
- Orbits of <0,2> acting on define E.
- Orbits of <0,1> acting on define F.

Orientable Map

- Theorem: A map is orientable if and only if the flag graph is bipartite.

Unique Embedding

- Theorem (Whitney): Each 3-connected planar graph admits a unique embedding in the sphere.
- Theorem (Mani). Let Aut G be the group of automorphism of a 3-connectede planar graph G and let Aut M be the group of automorphisms of the corresponding map. Then Aut G = Aut M.

Example - Exercises

- On the left there is an embedding of Q3 on torus.
- N1: Determine the rotation scheme for this embedding.
- N2: Determine the flag graph for this embedding.

Example - Exercises

- On the left there is a different embedding of Q3 on torus.
- N1: Determine the rotation scheme for this embedding.
- N2: Determine the flag graph for this embedding.
- .

Levi graph of a map

- Levi graph of a map M has the vertex set:
- VM t EM t FM,
- Edges are determined by the sides of flags (as triangles).
- WARNING: The graph on the left is not simple!!

Characterisation

- Theorem: Levi graph of a map is simple if neither 1-skeleton nor 1-co-skeleton has a loop.
- Definition: A map M is simple,if and only if its Levi graph is simple.

Homework

- H1: Given Flag graph of a map M, determine whether M is simple! (Prove previous theorem)

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