1. Maps

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

3. Cellular (or 2-cell) embedding Embedding  :G ! S is cellular (or 2-cell), if S \  (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 K 4 in torus S 1. The first one is cellular, the second ons is not!

4. 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.

5. Genus of a Graph Let  (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.

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

7. 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).

8. 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:  : S ! S : S !{-1,+1}. with the property: Permutation  acts cyclically on S[v], for each v 2 V. (s) =  (r(s)), for each s 2 S. [Hence is a voltage assignment. In our case: (s) = (r(s))]. The triple (G, , ) is a called a rotation scheme, defining a 2-cell embedding of G into some surface.

9. Interpretation of Rotation Scheme We follow arcs starting at s0 until we return to the initial arc. s à s0, s à  (s). positive à True. While s  s0 do  If positive then If (s) = 1 then  s à  (s) else  positive à False;  s à  (s) -1  else If (s) = 1 then  s à  (s) -1  else  positive à False;  s à  (s)  2 (s)  (s) sr(s)  3 (s)  4 (s)  (r(s))  2 (s)  (s) sr(s)  3 (s)  4 (s)  (r(s))

10. Rotation Scheme and Rotation Projection Rotation scheme can be represented by rotation projection. Rotation  can be reconstructed from the bottom drawing. Each arc s carries (s) = 1.

11. Example On the left we see the rotation projection of K 4. 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.

12. 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 = (e 1,e 2,..., e k ) such that (C) := (e 1 ) (e 2 )... (e k ) = -1

13. 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.

14. Let V,E,F be disjoint (finite) sets.  µ 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 ve f

15. 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.

16. 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.

17. 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 V e = {v 2 V| (v,e) 2 A}. Assume |V e | · 2, for each e 2 E. We may define r:A ! A by: r(v,e) = (w,e) if V e = {v,w} and r(v,e) = (v,e) if V e = {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 |V e | = 2.

18. 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.

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

20. 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 .

21. 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.

22. 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.]

23. Some limits of flag systems On the left we see K 4 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.

24. 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.

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

26. From flags to flag graph. First the vertices.

27. From flags to flag graph. First the vertices. Next: three kinds of new edges: along the edges across the edges. across the angles.

28. Flag graphs for 2-cell embedded graphs. Flag graph  is: - connected - trivalent - contains a 2-factor of form m C 4.

29. 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.

30. Yet another view to flag graphs. We may start with three involutions:  0,  1,  2 :  !   0 2 =  1 2 =  2 2 = 1, each fixed-poit free.  0  2 =  2  0, also fixed-point free. Each invoultion corresponds to a 1-factor. Together they define a cubic graph: the flag graph  (  ). The group, called monodromy group must act transitively on . [This is eaquivalent to saying that  (  ) is connected.] These axioms define a (combinatorial) map on a surface.

31. Combinatorial Map. Combinatorial map is defined by three involutions satisfying the axioms from the previous slide. Orbits of acting on  define V. Orbits of acting on  define E. Orbits of acting on  define F.

32. Orientable Map Theorem: A map is orientable if and only if the flag graph is bipartite.

33. 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.

34. Example - Exercises On the left there is an embedding of Q 3 on torus. N1: Determine the rotation scheme for this embedding. N2: Determine the flag graph for this embedding.

35. Example - Exercises On the left there is a different embedding of Q 3 on torus. N1: Determine the rotation scheme for this embedding. N2: Determine the flag graph for this embedding..

36. 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!!

37. 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.

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