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.
• gS ? G voltage assignment.
• Condition for each s 2 S we have
  gsgr(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), gs(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 graph q.

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 1s 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 lt 6 in H.
• Show that H is the smallest cubic graph with no
  cycles of length lt 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
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
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 lta,ban,b2, abba-1gt
• 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,?) lt (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
  minimal valence 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 2
  Zai
• Let S(H) (i,j,e,e,q)e and e of type i ! j,
  q 2 Zbij
• 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 ru,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 ru,B ?
  rv,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 ru,B rv,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-n1
  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 ?h 2 0,1m, ?h(e) 1,
  if e 2 C(h) and ?h(e) 0, represents C(h). The
  set of all charactersitic vectors spans a m-n1
  dimensional Z-module in Zm. This can be also
  viewed as a free abelian group isomorphic to
  Zm-n1.
• This group is called the first homology group
  H1(G,Z). We may replace Z by Zk and obtain the
  first Zk homology group Zkm-n1.

47
Pseudohomological Covers

• Idea Let G be a graph and T its spanning tree
  and with the edges H h1,h2,...,hm-n1
  E(G)\E(T). Let ?(H) be a group with m-n1
  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.

51
Homological Elementary Abelian Covers

• Let G be a graph with a spanning tree T. Let k
  m-n1 be the number of edges in G\T. Define the
  voltage assignment ? such that each non-tree edge
  gets the voltage ei (0,0,..,0,1,0,...,0) 2 Zpk.
• Claim If p is prime, then Cov(G,?) is
  independent of T.
• Question What happens in the case p is not prime?

52
Tree-To-Tree Switch

• Let T and T be two spanning trees of G. Let H
  h1, h2, ..., hk be the co-tree edges of T. Let
  r be the root of G. For each vertex w 2 V(G)
  there is a unique path P(T,w,r) on the three T
  from w to v. Let S(w) µ H be the collection of
  co-tree edges on this path. Let S(w) be the label
  given to w. Hence ?(w) ? hi hi 2 S(w).
• Claim Starting with homological voltage
  assignment relative to T and applying the
  tree-to-tree switch ?, the voltages are given as
  follows
• The edges on T get voltage 0.
• An edge e uv on a co-tree T get the voltage
• k(e) S(u) S(v) if e 2 T.
• k(e) S(u) S(v) h(e) if e ? T.
• Each co-tree edge e defines a cycle C(e). The net
  voltage on C(e) is equal to k(e).
• The voltages k(e), for e ? T span the whole Z2k.

53
Exercises

• N1. Let Znk be an elementary abelian group. Let S
  be a set of generators with the following
  property. Each element is a 0-1 vector. They
  generate the whole group.
• Show that S k.
• Show that there is an automorphism of the group
  mapping S to the standard generating set.

54
Real Homological Cover

• Let G be a graph with a given cycle basis C1, C2,
  ..., Ck. Direct each cycle and assign to each
  edge of Ci the voltage ei 2 Znk. The final
  voltage assignmnet is given by adding the partial
  voltages.
• An example is given on the left. The cycle basis
  is determined by a spanning tree.

(0,1)
(0,1)
(1,1)
(1,0)
Z22
(1,0)
55
Least Common Cover

• Theorem There exist finite connected graphs H1,
  H2, G1, G2 such that G1 and G2 are both double
  covers of H1 and H2.
• Proof. We start with graphs G G(5,2) and X
  that we know from earlier.

56
GX and G G

• Given two graphs G and H we form GH by adding an
  edge between them.
• On the left we see G X and G G.
• The resulting graph depends on the choice of the
  two vertices.

57
H1 and H2

• Define H1 and H2 as follows
• H1 G X X and H2 G G X.

58
Covers of GH.

• A double cover of GH can be split into two
  double covers G and H and then joint them by a
  pair of edges. We denote the resulting graph by
  G H.
• For instance KC(G X) KC(G) KC(X) G(10,3)
  G(10,3).

59
End of Proof

• Let G1 G(10,3) G(10,3) G(10,3) and G2
  G(10,3) G(10,3) 2X.
• G1 and G2 are distinct. They are both covers of
  H1 and H2.