Covering Graphs

1. Covering Graphs • Motivation: • Suppose you are taken to two different labyrinths. Is it possible to tell they are distinct just by walking around? • Let us call the first graph maze X, and the second one Y.

2. Question • Is it possible to distinguish between the two mazes? • Answer: Yes, we can. In the upper maze there are two adjacent trivalent vertices. This is not the case in the lower maze.

3. Local Isomorphism • On the other hand we cannot distinguish (locally) between the upper and lower graph. • To each walk upstairs we can associate a walk downstairs.

4. One More Example • C4 over C3 is no good. However, C6 over C3 is Ok.

5. Fibers and Sheets. • We say that C6 is a twosheeted cover over C3. Red vertices are in the same fiber. Similarly, the dotted lines belong to the saem fiber. • Graph mapping f: C6 C3 is called covering projection. • Preimage of a vertex f-1(v) (or an edge f-1(e)) is called a fiber. • The cardinality of a fiber is constant. k =|f-1(v)| is called the number of sheets.

6. One More Example • The cube graph Q3 is a two fold cover over complete graph K4. • The vertex fibers are composed of pairs of antipodal vertices.

7. Covers over Pregraphs • Graph K4 can be understood as a four-fold cover over a pregraph on one vertex (one loop and one half-edge).

8. Voltage Graphs • X = (V,S,i,r) – connected (pre)graph. • (G,A) – permutation group  acting on space A. • g:S G – voltage assignment. • Condition: for each s 2 S we have g[s]g[r(s)] = id.

9. Voltage Graph Determines a Covering Graph • Each voltage graph (X,G,A,g) determines a covering graph Y and the covering projection f: Y  X as follows: • Covering graph Y = (V(Y),S(Y),i,r) • V(Y) := V(X) x A • S(Y) := S(X) x A • i: S(Y)  V(Y): i(s,a) := (i(s),a). • r: S(Y)  S(Y): r(s,a) := (r(s), g[s](a)). • Covering projection f • f: V(Y)  V(X): f(x,a) := x. • f: S(Y)  S(X): f(s,a) := s. • Sometimes we denote the covering graph Y by Cov(X;).

10. (Rhetorical) Questions • “Different” voltage graphs may give rise to the “same” cover. What does it mean the “same” and how do we obtain all “different” voltage graphs? • The voltage graph is determined in essence by the abstract group. What is the role of permutation group? • How do we ensure that if X is connected then Y is connected, too?

11. Kronecker Cover • Let X be a graph. The canonical double cover or Kronecker cover: KC(X) is a twofold cover that is defined by a voltage graph that has nontrivial voltage from Z2 on each of its edges. It can also be described as the tensor product KC(X) = X £ K2.

12. Homework • H1: Prove that Kronecker cover is bipartite. • H2: Prove that generalized Petersen graph G(10,2) is a twofold cover over the Petersen graph G(5,2). • H3: Determine the Kronecker cover over G(5,2). • H4: Determine a Zn covering over the handcuff graph G(1,1), that is not a generalized Petersen graph G(n,r).

13. Regular Covers • Let Y be a cover over X. We are interested in fiber preserving elements of Aut Y (covering transformations). • Let Aut(Y,X) · Aut Y be the group of covering transformations. • The cover Y is regular, if Aut(Y,X) acts transitively on each fiber. • Regular covers are denoted by voltage graphs, where permutation group (G, A) acts regularly on itself by left or right translations: (G, G).

14. Exercises • N1: Prove that each double sheeted cover is regular. • N2: Find an example of a three sheeted cover that is not regular. • N3: Express the graph on the left as a 6-fold cover over a pregraph on a single vertex.

15. Dipole qn • Dipole qn has two vertices joined by n parallel edges. We may call one vertex black, the other white. On the left we see q5. • Each dipole is bipartite, that is why each cover over n is bipartite too. Dipole q3 jeis cubic, sometimes called the theta graphq.

16. Cyclic cover over a dipole – Haar graph H(n). • H(37) is determined by number 37, actually by its binary representation (1 0 0 1 0 1). • k = 6 is the length of the sequence, hence group Z6. • (0 1 2 3 4 5) – positions of “1”. • Positions of “1”s: 0, 3 in 5. {0,3,5} are the voltages on . The corresponding covering graph is H(37). 0 3 5 Z6

17. Exercises • Graph on the left is called the Heawood graph H. Prove: • H is bipartite. • H is a Haar graph (Determine n, such that H = H(n)) • Express H as a cyclic cover over . • Show that there are no cycles of lenght < 6 in H. • Show that H is the smallest cubic graph with no cycles of length < 6.

18. Cages as Covering Graphs • A g-cage is a cubic graph of girth g that has the least number of vertices. • Small cages can be readily described as covering graphs.

19. 1-Cage • Usually we consider only simple graphs. For our purposes it makes sense to define also a 1-cage as a pregraph on the left. • 1-cage is the unique smallest cubic pregraph.

20. 2-Cage • The only 2-cage is the  graph. • We may view 2-cage, as the Kronecker cover over 1-cage. 1 1 Z2

21. K4, the 3-cage 3 2 • K4 is a Z4 covering over the 1-cage. • In general, we obtain a Z2n covering over the 1-cage by assigning voltage 1 to the loop and voltage n to the half-edge. • Exercise: What is the covering graph in such a case? 1 0 2 1 Z4

22. K3,3, the 4-cage 5 4 • K3,3 is a Z6 covering over the 1-cage. • It can also be seen as a Z3 covering over the 2-cage . • Exercise: Express K3,3 as a covering graph over . Dtermine a natural number n, such that K3,3 is a Haar graph H(n). 3 2 1 0 3 1 Z6

23. The Handcuff Graph G(1,1) • By changing the voltage on the loop of the 1-cage we obtain a double cover G(1,1), the smallest generalized Petersen graph, known as the Handcuff graph. 1 0 Z2

24. 0 i Zn j I graphs I(n,i,j) and Generalized Petersen graphs G(n,k) • Cyclic covers over the handcuff graph are called I-graphs. Each I-graph can be described by three parameters I(n,i,j) with i · j. In case i = 1 we call I(n,i,k) = G(n,k), the generalized Petersen graph. • In particular, I(5,1,2) is the 5-cage.

25. b a D7 The 6-cage • The 6-cage is the Heawood graph on 14 vertices. It is a 7-fold cyclic cover over the  graph. But it is also a dihedral cover over the 1-cage. • Let the presentaion of Dn be given as follows: Dn = <a,b|an,b2, ab=ba-1> • Then the Heawood is a covering described on the left.

26. Exercises • N1. Express the 7-cage as a covering graph. • N2. Express the 8-cage as a covering graph.

27. (3,1)-trees • A (3,1)-tree is a tree whose vertices have valence 3 and 1 only. • On the left we see the smallest (3,1)-trees I,Y and H.

28. (3,1)-cubic graphs • A (3,1)-cubic graph is obtained from a (3,1)-tree by adding a loop at each vertex of valence 1. • On the left we see the smallest (3,1)-cubic graphs I(1,1,1),Y(1,1,1,1) and H(1,1,1,1,1).

29. Coverings over (3,1)-cubic graphs Zn j i • By putting 0 on the tree edges and appropriate voltages on the loops of (3,1)-cubic graph we obtain their Zn coverings. • In the case of the graphs on the left we obtain the I-graphs, Y-graphs and H-graphs: I(n,i,j),Y(n,i,j,k) and H(n,i,j,k,l). j i k k i j l

30. Covers Determined by Graphs • We know already that there exists a cover, namely Kronecker cover, that depends only on X itself and the voltage assignment plays a minor role. • Now we will present some covers that have a similar property.

31. Coverings and Trees • Let X be a connected graph and let Cov(X) denote all connected covers over X: • Cov(X) = {(Y,)| Y connected and : Y ! X, covering projection}. For each connected X we have (X,id) 2 Cov(X). • Proposition: For a connected X we have Cov(X) = {(X,id)} if and only if X is a tree. • This fact holds both for finite and locally finite trees.

32. Universal cover • Let X, Y and Z be connected graphs and let : Y ! X and :Z ! Y be covering projections. • On the other hand, we may consider the class Cov(X) of all coverings over X. We may introduce a partial order in Cov(X). (Y,) < (Z,) if there exists a covering projection (Z,) 2 Cov(Y) so that  = . • Proposition: Any connected finite or locally finite graph X can be covered by some tree T; : T ! X. • Proposition: Any connected finite or locally finite graph X can be covered by at most one tree T. • Proposition: Let : T ! X be a covering projection form a tree to a connected graph X. Then for each covering : Y ! X there exists a covering : T ! Y such that  = . • Corollary: For each connected X the poset Cov(X) has a maximal element (T,) where T is a tree. • The maximal element (T,) 2 Cov(X) is called the universal covering of X.

33. Construction of Universal Cover • There is a simple construction of the universal covering projection. • Let X be a connected graph and let T µ X be a spanning tree. Furthermore, let S = E(G) \ E(T) be the set of edges not in tree T. • Consider S to be the set of generators for a free group F(S) and F(S) to be the voltage group. • Let us assing voltages on E(G) as follows: • If e 2 E(T) the voltage on e is identity. • If e 2 S the voltage is the corresponding generator (or its inverse) • Note: The construction does not depend on the choice of direction of edges. • Proposition: The described construction gives rise to the universal cover.

34. Examples • Example: The universal cover over any regular k-valent graph is a regular infinte tree T(1,k).

35. Valence Partition and Valence Refinement • Let G be a graph and let B = {B1, ..., Bk} be a partition of its vertex set V(G) for which there are constants rij, 1 · i,j · k such that for each v 2 Bi there are rij edges linking v to the vertices in Bj. Let R = [rij] be the corresponding k £ k matrix, Then B is called valence partition and R is called valence refinement. If k is minimal, then B is called minimalvalence partition and R is called minimal valence refinement. • Two refinements R and R’ are considered the same if one can be transformed to the other one by simultaneous permutation of rows and columns. • A refinement is uniform, if each row is constant.

36. Construction • Given graphs G and G’ with a common refinement. • Let mij denote the number of arcs in G of type i ! j. • Let ni denote the number of vertices in G of type i. • Let bij = lcm(mij)/mij. (If mij = 0 , let bij undefined). • Let ai = lcm(mij)/ni. • Note that bij and ai depend only on the common matrix R and are the same for both graphs G and G’. • Let l(e) or l(e’) be a linear order given to all type i ! j arcs with a common initial vertex i(e) (or i(e’)). • Let V(H) = {(i,v,v’,p)|v and v’ of type i, p 2Zai} • Let S(H) = {(i,j,e,e’,q)|e and e’ of type i ! j, q 2Zbij} • r(i,j,e,e’,q) := (j,i,r(e),r(e’),q) • i(i,j,e,e’,q) := (i,i(e),i(e’),q rij + l(e)-l(e’)} • H is a common cover of G and G’.

37. Computing Minimal Valence Refinement • Let r[u,B] denote the number of edges linking u to the vertices in B. • Algorithm [F.T.Leighton, Finite Common Coverings of Graphs, JCT(B) 33 1982, 231-238.] • Step 1. Place two vertices in the same block if and only if they have the same valence. • Step 2. While there exist two blocks B and B’ and two distinct vertices u,v in B with r[u,B’]  r[v,B’] repeat the following: • Partition the block B into subblocks in such a way that two vertices u,b of B remain in the same block if and only if r[u,B’] = r[v,B’] for each B’ of the previous partition. • Step 3. From minimal valence partition B compute the minimal vertex refinement R. • Note: We may maintain R during the run of the algorithm as a matrix whose elements are sets of numbers.

38. Comon Cover • Theorem. Given any two finite graphs G and H, the following statements are equivalent: • G and H have the same universal cover, • G and H have a common finite cover, • G and H have a common cover, • G and H have the same minimal valence refinement. • G and H have the same some valence refinement. • Homework. Find the result in the literature and construct a finite comon cover of G(5,2) and G(6,2).

39. Petersen graph • An unusual drawing of Petersen graph.

40. Petersen graph G(5,2) and graph X.

41. Kronecker Cover - Revisited • Kronecker cover KC(G) is an example of covers, determined by the graph itself. • Exercise. Show that G(5,2) and X have the same Kronecker cover.

42. THE covering graph • Let G be a graph with the vertex set V. By THE(G) we denote the following covering graph. • To each edge e = uv we assing transposition e = (u,v) 2 Sym(V). The resulting covering graph has two components, one being isomorphic to G. The other componet is called THE covering graph.

43. Examples • On the left we see The covering graph of K2,2,2. • The construction resembles truncation. • Each vertex is truncated and an inverse figure is placed in the space provided for it. • Theorem: If G is planar, then THE(G) is planar.

44. Homework • H1. Given connected graph G with n vertices and e edges and with valence sequence (d1, d2, ..., dn). Determine the parameters for THE(G). • H2. Determine all connected graphs G for which girth(G)  girth(THE(G)).

45. The fundamental group of a graph. • Let G be a connected graph rooted at r 2 V(G) and let  denote the collection of closed walks rooted at r. • Let  and  be two closed walks rooted at r. The compositum  is also a closed walk rooted at r. • We may also define -1 as the inverse walk. • Finally, we need equality (equivalence). • 12 ~ 1 e e-12. • (G,r) := /~ is a group, called the fundamental group of G (first homotopy group). • Fact: (G,r) is a free group generated with m-n+1 generators.

46. The first Homology group of a graph • Let G be a connected graph and T one of its spanning trees. Each edge h 2 G\T of the co-tree defines a unique cycle C(h) µ E(G). • The charactersitic vector h2 {0,1}m, h(e) = 1, if e 2 C(h) and h(e) = 0, represents C(h). The set of all charactersitic vectors spans a m-n+1 dimensional Z-module in Zm. This can be also viewed as a free abelian group isomorphic to Zm-n+1. • This group is called the first homology group H1(G,Z). We may replace Z by Zk and obtain the first Zk homology groupZkm-n+1.

47. Pseudohomological Covers • Idea: Let G be a graph and T its spanning tree and with the edges H = {h1,h2,...,hm-n+1} = E(G)\E(T). Let (H) be a group with m-n+1 interchangeable generators H. The pseudohomological-cover HOM(G,,T) is determined by a voltage graph with (e) = id, for e 2 E(T) and (h) = h, for h 2 E(G)\E(T). • Main Question. Is HOM(G,,T) independent of the choice of T and the selection of the generators or their inverses? If the answer is yes, the covering is called homological cover.

48. Pseudohomological 2-cover • Let G be a graph and T its spanning tree.The pseudohomological 2-cover HOM(G,Z2,T) is determined by a voltage graph with (e) = 0, for e 2 E(T) and (e) = 1, for e  E(T). • Theorem. If G is connected then HOM(G,Z2,T) is connected if and only if G is not a tree.

49. Example 0 • The two voltage graphs on the left determine different pseudohomological Z2 covers. • Cov(G,2) is bipartite and Cov(G,1) is not. 1 0 0 1 1 Z2 0 0 1 1 2 0

50. Switching • Let (G,) be a voltage graph. Let : V(G) ! be an arbitrary mapping, called switching, that assigns voltages to vertices. Define a new voltage assignment  as follows: • (s) := (i(s))  (s) (i(r(s))-1. •  is well-defined. • Namely (r(s)) = (i(r(s))) (r(s)) (i(s))-1. • Hence (r(s))-1 = (i(s)) (r(s))-1(i(r(s)))-1 = (i(s)) (s) (i(r(s)))-1 = (s). • Clearly for any switching  the graphs Cov(G,) and Cov(G,) coincide. • Given (G,) and any spanning tree T. There exists a switching  such that the resulting e is identity on T. • If, in addition, T is rooted at v, we may select (v) = id (or arbitrarily) and this determines switching completely.