Table Of ContentPARTITION FUNCTIONS AND A GENERALIZED
COLORING-FLOW DUALITY FOR EMBEDDED GRAPHS
Bart Litjens∗and Bart Sevenster †‡
Abstract. Let G be a finite group and χ : G → C a class function. Let H = (V,E) be a directed
graphwithforeachvertexacyclicorderoftheedgesincidenttoit. Thecyclicordersgiveacollection
F of faces of H. Define the partition function P (H) := χ(κ(δ(v))), where κ(δ(v))
7 χ κ:E→G v∈V
1 denotesthe productofthe κ-valuesofthe edgesincidentwith v (inorder),wherethe inverseistaken
P Q
0
for any edge leaving v. Write χ = m χ , where the sum runs over irreducible representations
2 λ λ λ
n λP o(Hf G)=w|iGth||Ec|haracχter(1χ)|λF|−an|Ed|mw|Vit|hP, wmhλere∈1Cisftohreeivdeernytitλy.elIefmHentisofcoGn.nAecmteodn,gitthiescporroovlleadrietsh,aat
a χ λ λ λ
J formula for the number of nowhere-identity G-flows on H is derived, generalizing a result of Tutte.
P
2 We show that these flows correspondbijectively to certain proper G-colorings of a covering graph of
the dual graph of H. This correspondence generalizes coloring-flow duality for planar graphs.
]
O
C Key words: embedded graph, coloring-flow duality, representation, partition function
. MSC 2010: 05C10, 05C15, 05C21, 20C15, 57M10
h
t
a
m 1 Introduction
[
1 Let H = (V,E) be a directed graph (not necessarily simple) with for each vertex v a cyclic
v order φ of the edges incident to v. For a vertex v, let d(v) denote the total degree of v.
v
0 If v is a vertex that is not isolated, we fix an edge e1 adjacent to v and let ei+1 denote the
2 v v
4 edge incident to v that in the cyclic order of v comes after eiv, for i = 1,...,d(v) −1. For
0 v ∈ V, let σ : {e1,...,ed(v)} → {±1} be the function that takes the value −1 on edges that
0 v v v
. are directed outwards at v and 1 on the edges that are directed inwards at v. A face of H
1
is given by a walk such that for every edge e in the walk, the next edge in the walk is the
0
7 edge that is subsequent in the cyclic order at the end vertex of e.
1 Let G be a finite group and χ : G → C a class function of G, i.e., a function that is
:
v constant on conjugacy classes of G. Define the partition function P with respect to χ and
χ
i
X evaluated at H to be
r
a (1) P (H):= χ(κ(φσv)),
χ v
κ:E→Gv∈V
X Y
where κ(φσvv) := id=(v1)κ(eiv)σv(eiv). Note that the partition function is well-defined and does
not depend on the direction of the edges. It is a standard fact from representation theory
Q
that one can write χ = m χ , where the sum runs over irreducible representations λ of
λ λ λ
G with character χ and with m ∈ C for every λ.
λ λ
P
In this paper we give a formula of the partition function (1) and study this formula in
∗University of Amsterdam. Email: [email protected]
†University of Amsterdam. Email: [email protected]
‡Korteweg-De Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands.
The research leading to these results has received funding from the European Research Council under the
European Union’s SeventhFramework Programme (FP7/2007-2013) / ERCgrant agreement n◦ 339109.
1
some special cases. In Section 2.2 it will be proved that if H is connected, it holds that
(2) P (H) = |G||E| χ (1)|F|−|E|m|V|,
χ λ λ
λ
X
where F denotes the collection of faces of H and 1 is the identity element of G. For every
λ, the coefficient χ (1) equals the dimension of the representation λ.
λ
In Section 3.2 formula (2) will be studied for a particular class function and graph.
In this way, another proof of a special case of the Generalized Frobenius Formula [11] is
found. If k and g are nonnegative integers, the Generalized Frobenius Formula counts a
certain number of homomorphisms from the fundamental group of a k-fold punctured closed
orientable surface of genus g into the given group G.
When χ equals the character of the regular representation of G, the partition function
P (H) counts the number of G-flows on H. Using inclusion-exclusion, a formula that counts
χ
the number of nowhere-identity G-flows on H is derived in Section 3.4. Recently, this result
was independently found in [3]. As the authors of this paper also mention, the case where
G is nonabelian seems not to have been discussed extensively yet.
In Section 4 a theorem of Heffter-Edmonds-Ringel is used to cellularly embed H in
a closed orientable surface S. It is proved that nowhere-identity G-flows on H correspond
bijectively to certain proper G-colorings of a (finite) covering graph of H∗, the dual graph
of H on S. This correspondence generalizes the coloring-flow duality for planar graphs, that
was first proved by Tutte in [10]. Our generalization is different from the one discussed in
[1], where only R-valued flows are considered.
2 The partition function
2.1 Preliminaries on representation theory
In this section we review some of the representation theory of finite groups. More details
and proofs of the results stated may be found in the first two chapters of [8]. Furthermore,
some notation is introduced that is used in Sections 2.2 and 3.4.
Let G be a finite group. A representation of G is a homomorphism ρ : G → GL(V),
where V is a finite-dimensional C-vector space, called the representation space, and GL(V)
is the group of invertible linear maps from V to itself. Identify GL(V) with the group of
invertible square matrices of size dim(V). In this paper all representations ρ are assumed to
be unitary, i.e., ρ(g) is a unitary matrix for all g ∈ G.
If ρ and ψ are representations of G with representation spaces V and W respectively,
then a G-equivariant map from V to W is a linear map φ : V → W such that φ(ρ(g)·v) =
ψ(g)·φ(v) for all g ∈ G and v ∈ V. The representations ρ and ψ are isomorphic if there
exists a bijective G-equivariant map from V to W.
A representation ρ : G → GL(V) is called irreducible if there is no nontrivial subspace
U of V such that ρ(g) · U = U for all g ∈ G. Every representation is isomorphic to a
direct sum of irreducible representations. The number of pairwise nonisomorphic irreducible
representations of G equals the number of conjugacy classes of G.
Associated to a representation ρ is the function χ : G→ C, called the character, that is
definedbyχ(g) := tr(ρ(g)), wheretrstandsforthetrace. Tworepresentationsareisomorphic
if and only if their characters agree. Thecharacter of a representation is a class function and
the complex space of class functions is equipped with an inner product h·,·i defined by
1
hχ ,χ i := χ (g)χ (g−1),
1 2 1 2
|G|
g∈G
X
2
for class functions χ and χ . The characters of the irreducible representations form an or-
1 2
thogonal basis with respect to this inner product. Of crucial importance is the orthogonality
relation below. In the following, let δ denote the Kronecker delta function that is 1 if a = b
ab
or a is equivalent to b and zero elsewise.
Theorem 2.1 (Schur orthogonality). Let ρ and ψ be irreducible (unitary) representations
of G with characters χ and χ , then
ρ ψ
|G|
ρ(g−1) ·ψ(g) = δ δ δ ,
n,m n′,m′ nm′ mn′ ρψ
χ (1)
ρ
g∈G
X
for all indices n,m,n′,m′ = 1,...,χ (1).
ρ
It can be proved that the dimension χ (1) divides |G|.
ρ
An important representation is the regular representation ρ . Let C[G] be the group
reg
algebra of G. It is the vector space over C with basis G. The group multiplication defines
an algebra structure on C[G]. Then the regular representation is defined by ρ : G →
reg
GL(C[G]),g 7→ L ,where L is the map that denotes left multiplication by g in C[G]. Its
g g
character χ is |G| on the identity element of G and is zero elsewhere. It decomposes as
reg
χ (1)χ , where the sum runs over all irreducible representations λ of G.
λ λ λ
P
2.2 A formula
In this section we derive a formula for the partition function that is defined in (1). It is
a direct computation that relies mostly on Schur orthogonality and the observation that
the faces determine the matrix coefficients that give nonzero contributions. The proof is
essentially the same as in [3].
With notation andunderthesame assumptionsas in thebeginningof theintroduction,
we have that
Theorem 2.2. If H is connected, the partition function P (H) equals
χ
P (H) = |G||E| χ (1)|F|−|E|m|V|,
χ λ λ
λ
X
where the sum runs over all irreducible representations λ of G.
Proof. Let Λ:= {λ | m 6= 0}, then we calculate
λ
P (H) = χ(κ(φσv))
χ v
κ:E→Gv∈V
X Y
= m χ (κ(φσv))
λ λ v
κ:E→Gv∈V λ
X Y X
(3) = m χ (κ(φσv)).
µ(v) µ(v) v
κ:E→Gµ:V→Λv∈V
X X Y
For a fixed µ :V → Λ, we get a contribution
d(v)
m χ (κ(φσv)) = m χ ( κ(ei)σv(eiv))
µ(v) µ(v) v µ(v) µ(v) v
κ:E→Gv∈V κ:E→Gv∈V i=1
X Y X Y Y
d(v)
= m tr( µ(v)(κ(ei)σv(eiv)))
µ(v) v
κ:E→Gv∈V i=1
X Y Y
χµ(v)(1) d(v)
(4) = mµ(v) (µ(v)(κ(eiv)σv(eiv)))ji,ji+1,
κ:XE→GvY∈V j1,...X,jd(v)=1Yi=1
3
where we used that the representation µ(v) is a homomorphism in the second equality,
expanded the trace of a product of matrices in the third equality and where the subindices
i of j are taken modulo d(v) for all i.
i
Let e ∈ E be an edge that is directed from u to v. Then in expression (4) there is a
term of the form
χµ(u)(1)χµ(v)(1)
(5) (µ(u)(κ(e)−1)) ·(µ(v)(κ(e))) .
a,b c,d
a,b=1 c,d=1 κ:{e}→G
X X X
Hence, by Theorem 2.1 and by connectivity of H, we can assume that µ : V → Λ is a
constant function, as otherwise expression (5) is zero.
LetF beaface ofH withfacial walk W consistingof theedges e ,...,e (inthis order)
F 1 r
for some r ≥ 1. We may assume that e is directed from v to v for i = 1,...,r −1 and
i i i+1
that e is directed from v to v . Fix a function µ : V → Λ such that µ(v) = λ for all v ∈ V.
r r 1
In expression (4) the following term occurs
χλ(1) r
(6) (λ(κ(e )−1)) (λ(κ(e ))) ,
i j4i,j4i+1 i j4i+2,j4i+3
j1,..X.,j4r=1κ:{e1,X...,er}→GYi=1
where the subindices i of j are taken modulo 4r for all i. Since F is a facial walk, we have
i
that j = j for all i. If the term (6) is nonzero, by Theorem 2.1 we must have that
4i 4i−1
j = j and j = j for all i. Hence there is an index that is dominant, i.e., that
4i 4i+3 4i+1 4i+2
occurs in every matrix coefficient in the above product.
Applying the above for every face yields the number of faces many indices that range
between 1 and χ (1) for every λ. Hence the faces determine the matrix coefficients that
λ
give nonzero contributions to the partition function. For every edge e, we need only to
consider the matrix coefficient whose indices correspond to the faces on either side of e.
Using expressions (3) and (4) we compute
P (H) = m χ (κ(φσv))
χ µ(v) µ(v) v
κ:E→Gµ:V→Λv∈V
X X Y
= m χ (κ(φσv))
λ λ v
λ κ:E→Gv∈V
X X Y
= m|V|χ (1)|F| (λ(κ(e)−1)) (λ(κ(e)))
λ λ 1,1 1,1
λ κ:E→Ge∈E
X X Y
= m|V|χ (1)|F| (λ(κ(e)−1)) (λ(κ(e)))
λ λ 1,1 1,1
λ e∈Eκ:{e}→G
X Y X
= |G||E| χ (1)|F|−|E|m|V|,
λ λ
λ
X
whereinthethirdequality weusedthatevery choice ofindicesbetween1andχ (1)gives the
λ
same contribution by Theorem 2.1 and in the last equality Theorem 2.1 is used again.
3 Consequences of Theorem 2.2
3.1 Preliminaries on surfaces and the fundamental group
We assume some familiarity with the theory of surfaces and the fundamental group. A brief
review is given in this section. For more information on surfaces and the fundamental group
we refer to [4] and [5] respectively.
4
A surface is a connected topological space that is Hausdorff and such that every point
has an open neighborhood that is homeomorphic to an open disc in R2. A closed surface
is a surface that is compact. If a surface admits a triangulation by oriented triangles such
that for any edge of a triangle on the surface the orientations of the neighboring triangles
are opposite, it is called orientable. A surface is oriented if such a triangulation is given. A
closed orientable surface is homeomorphic to the sphere with g handles attached to it, for
some g ≥ 0 which is called the genus.
Let X be a topological space. A loop is a continuous function f from the unit interval
[0,1] in R to X such that f(0) = f(1). If x ∈ X, then a loop f is based at x if f(0) =
f(1) = x. Two loops f and f are said to be homotopic if there exists a continuous function
0 1
F : [0,1]×[0,1] → X such that F(t,0) =f and F(t,1) = f for all t ∈ [0,1].
0 1
Fix x ∈ X. Beinghomotopic definesanequivalence relation onthesetof loops basedat
x. Theset of equivalence classes forms a group with respect to concatenation of loops, called
the fundamental group of X (based at x) and denoted by π (X,x). If X is path-connected
1
then π (X,x) and π (X,y) are isomorphic for y ∈ X, which justifies the notation π (X).
1 1 1
3.2 Counting homomorphisms
In this section a first consequence of Theorem 2.2 is presented. Fix a g ≥ 0 and let S
g
be the closed orientable surface of genus g. Fix a point x ∈ S and remove it from S .
g g
The remaining surface is denoted S and is called the punctured surface of genus g. Its
g,1
fundamental group can be described by a set of generators subject to one relation
g
(7) π (S ) =ha ,...,a ,b ,...,b ,c | [a ,b ] = c−1i,
1 g,1 1 g 1 g i i
i=1
Y
where [x,y] = xyx−1y−1 denotes the commutator of x and y.
Let G be a finite group with identity element 1 and let C be a conjugacy class of G.
We define
g
A (G,C) := |{(x ,...,x ,y ,...,y ,z) ∈ G2g ×C | [x ,y ]= z−1}|.
g 1 g 1 g i i
i=1
Y
Then we have that
A (G,C) = |{φ :π (S )→ G | φ(c) ∈ C}|,
g 1 g,1
where the φ that are counted are assumed to be group homomorphisms and c is one of the
generators in the presentation (7) of π (S ).
1 g,1
The parameter A (G,C) has a geometric interpretation in terms of coverings of S ,
g g,1
see [11]. The following theorem, that is a special case of the Generalized Frobenius Formula
[11], is derived from Theorem 2.2.
Theorem 3.1. With notation as above, it holds that
χ (C)
A (G,C) = |G|2g−1|C| λ ,
g χ (1)2g−1
λ
λ
X
where the sum runs over all irreducible λ and where χ (C)= χ (h) for an h ∈ C.
λ λ
Proof. Consider the following function
|G|
if h ∈ C
f :G → C, f (h) = |C|
C C
(0 otherwise.
5
It is clear that f is a class function. Write f = m χ . Then
C C λ λ λ
m = 1 f (g)χ (gP−1) = χ (C−1),
λ C λ λ
|G|
g∈G
X
where C−1 is the conjugacy class consisting of the inverse elements of C. Let H be a graph
with one vertex and 2g loops attached to it in such a way that for odd i, the i-th loop only
intersects the (i+1)-th loop (once). This graph can be embedded on S . Theorem 2.2 gives
g
(8) P (H) = |G|2g χ (1)1−2gχ (C−1).
fC λ λ
λ
X
On the other hand, by definition of the partition function we have
g
|G|
P (H) = ·|{(x ,...,x ,y ,...,y ) ∈ G2g | [x ,y ] ∈C}|
fC |C| 1 g 1 g i i
i=1
Y
g
|G|
= ·|{(x ,...,x ,y ,...,y ,z) ∈G2g ×C−1 | [x ,y ]= z−1}|
1 g 1 g i i
|C|
i=1
Y
|G|
(9) = A (G,C−1).
g
|C|
Setting C := C−1 and comparing equations (8) and (9) yields the theorem.
3.3 Preliminaries on graph embeddings
Inthis section we give some preliminaries on graphembeddings. Thesepreliminaries andthe
notation are used in Sections 3.4 and 4.2. More information concerning graph embeddings
may be found in [4].
Let H be a (undirected) graph and S a closed orientable surface. An embedding of H
into S, which is denoted by a map i :H → S, is an assignment of points and simple arcs in S
to vertices and edges in H respectively, satisfying the following conditions: distinct vertices
are associated with distinct points. A simple arc representing an edge has as its endpoints
the points assigned to the vertices of the correspondingedge, and no other points on the arc.
Lastly, the interiors of distinct simple arcs should be disjoint.
Let i :H → S be an embedding. Then it is called a cellular (or 2-cell) embedding if all
connected components of S\i(H) are homeomorphic to an open disc. There is the following
theorem, that is often attributed to Edmonds [2], Heffter [6] and Ringel [7].
Theorem 3.2. Let i : H → S be a cellular embedding. If S is oriented then i induces for
every vertex v of H a cyclic order of the edges incident to v. Conversely, if we are given a
connected graph H then giving a cyclic order for every vertex defines a cellular embedding of
H in a closed oriented surface.
Let H bea connected graph with a cyclic order for every vertex. We definethe number
g(H) to be the genus of the closed orientable surface in which H cellularly embeds by
Theorem 3.2. This embedding gives a collection F of faces of H. It holds that
(10) |V|−|E|+|F|= 2−2g(H).
The number |V|−|E|+|F| is called the Euler characteristic of H.
Given a cellular embedding i of H into a closed orientable surface S, the dual (graph)
H∗ of H in S can be constructed. Its vertices correspond to faces of H and there is an edge
between two vertices in H∗ if the corresponding faces in H share an edge. The embedding i
gives rise to a map i∗ that embeds H∗ in S, called the dual embedding.
6
3.4 Counting nowhere-identity flows
Let G be a finite group with identity element 1. Let H = (V,E) be a connected directed
graph with for each vertex v a cyclic order φ of the edges incident to v. For each v ∈ V, let
v
e1 beanedgeincidenttov andletei+1 betheedgethatcomes afterei inthecyclic order,for
v v v
i = 1,...,d(v)−1, whered(v) is thetotal degree of v. AG-flow on H is afunction ψ :E → G
such that d(v)ψ(ei)σv(eiv) = 1 (in G) for all v ∈ V. A G-flow that never evaluates to the
i=1 v
identity is called a nowhere-identity G-flow.
Q
If G is abelian, Tutte proved in [10] that the number of nowhere-zero G-flows depends
only on the size of G. In this section we generalize Tutte’s result by deriving a formula that
counts the number of nowhere-identity G-flows, where G is any finite group. This number
depends only on the multiset consisting of the dimensions of the irreducible representations
of G, a result that recently was found independently by the authors of [3]. We start with
enumerating G-flows on H.
Let χ be the character of the regular representation of G. With notation as in the
reg
beginning of the introduction, the partition function is computed to be
(11) P (H) = χ (κ(φσv)) = |G||V|·|{κ :E → G |κ(φσv)= 1 for all v}|.
χreg reg v v
κ:E→Gv∈V
X Y
The condition that κ(φσv) = 1 for all v, is the conditon for κ to be a G-flow on H. Hence
v
P (H) equals |G||V times the number of G-flows on H, which is denoted by N (H). On
χreg G
the other hand, as m = χ (1) for the character of the regular representation, Theorem 2.2
λ λ
reads
(12) P (H) = |G||E| χ (1)|V|−|E|+|F| = |G||E| χ (1)2−2g(H),
χreg λ λ
λ λ
X X
where equation (10) is used in the second equality. Putting together equations (11) and (12)
yields
N (H) = |G||E|−|V| χ (1)2−2g(H),
G λ
λ
X
for the number of G-flows on H.
Next we consider nowhere-identity flows on H with values in G. If A ⊂ E is a subset of
the edge set of H, let (V,A) denote the subgraphof H where the edges in E\A are removed.
Let χ be a class function of G. Write
P (H) = P ((V,E \B)),
χ χ
B⊂E
X
where e
P (H)= χ(κ(φσv)).
χ v
κ:E→G\{1}v∈V
X Y
Inclusion-exclusion gives e
(13) P (H) = (−1)|E\B|P ((V,B)) = (−1)|E\B| P (C),
χ χ χ
B⊂E B⊂E C
X X Y
e
where the product in the last expression runs over components C of the subgraph (V,B) of
H. In the second equality we used that the partition function is multiplicative with respect
to the disjoint union of graphs. Let V ,E and F denote the vertex set, edge set and
C C C
collection of faces of C respectively. Then equation (13) together with Theorem 2.2 gives
(14) P (H) = (−1)|E\B||G||B| χ (1)|FC|−|EC|m|VC|,
χ λ λ
B⊂E C λ
X YX
where we used thaet |G||EC| = |G|PC|EC| = |G||B|.
C
Q
7
Theorem 3.3. The number N (H) of nowhere-identity G-flows on H is given by
G
NG(H) = e (−1)|E\B||G||B|−|V| χλ(1)2−2g(C),
B⊂E C λ
X YX
e
where the product runsoverallcomponents C ofthe subgraph (V,B)ofH andwhere 2−2g(C)
denotes the Euler characteristic of C.
Proof. Let χ be the character of the regular representation of G. The observation that
reg
P (H) = |G||V|·N (H)
χreg G
combined with equation (14) for χ yields the theorem.
ereg e
Corollary 3.4. The number N (H) of nowhere-identity G-flows on H depends only on H
G
and the multiset consisting of dimensions of irreducible representations of G.
e
Proof. This follows directly from Theorem 3.3.
Thedihedralgroupof order8andthequaterniongrouparethesmallest exampleoftwo
nonisomorphicgroupswhosemultisets consistingofdimensionsofirreduciblerepresentations
coincide. Note that Corollary 3.4 incorporates the dependence on the size of the group, as
|G| = χ (1)2, where the sum runs over all irreducible representations λ of G. If G is
λ λ
abelian then χ (1) = 1 for every λ, of which there are |G| many. Hence in this case Theorem
λ
P
3.3 reduces to
N (H) = (−1)|E\B||G||B|−|V|+c(B),
G
B⊂E
X
e
where c(B) denotes the number of connected components of the subgraph (V,B) of H.
4 Coloring-flow duality for embedded graphs
4.1 Preliminaries on coverings
Inthissection somepreliminariesoncoverings aregiventhat, together withthepreliminaries
ongraphembeddingsinSection3.3,areusedtoprovethedualitybetweenflowsandcolorings
for embedded graphs in Section 4.2. For more background information the reader is referred
to [4]. Details concerning the universal covering graph may be found in [9]. Moreover, this
section serves as a setup of notation that is used in Section 4.2.
Let K and H be graphs. A homomorphism between K and H is a mapping from the
vertex set of K to the vertex set of H such that if two vertices are adjacent in K, their
images are adjacent in H. Let s : K → H be a homomorphism. Then s is said to be a
covering (of H) if it is surjective, and the (graph theoretic) neighborhood of each vertex of
K is in bijection with the (graph theoretic) neighborhood of its image in H under s. If there
is a loop at a vertex, then the vertex itself is in its own neighborhood.
For every connected graph H that is not a tree there exists a covering t : T → H with
the following properties: T is an infinite tree and if r :K → H is another covering of H with
K connected, then there exists a covering s : T → K such that r ◦s = t. The graph T is
called the universal covering graph of H and is unique up to isomorphism. If H is a tree, its
universal covering graph is defined to be H itself.
After choosing a vertex v of H, the vertices of T can be described as non-backtracking
walks in H starting at v. A vertex w in T is adjacent to the vertices that correspond to
simple extensions of the walk in H that is associated with w. The covering t : T → H then
is given by sending a vertex in T to the endpoint of the walk in H it corresponds to.
8
4.2 Coloring-flow duality
In this section we prove that flows on graphs that are embedded on a closed orientable
surface correspond bijectively to specific coverings of the dual of that graph. By doing
so, we generalize Tutte’s coloring-flow duality for planar graphs. We use the notation and
terminology as introduced in Sections 3.3 and 4.1. First we need to define the concept of a
local and global tension.
As in the previous sections, let G be a finite group with identity element 1 and let H =
(V,E) be a connected directed graph with a cyclic order at each vertex. If W = (e ,...,e )
1 n
is a walk in H for some n ≥ 1, then for a map ψ : E → G we define the height of W as
hψ(W) = ni=1ψ(ei)σW(ei), where σW(ei) = +1 if ei is traversed in the direction of ei and
−1 else. A local G-tension on H is a function ψ : E → G, such that h (W) = 1 for every
ψ
Q
facial walk W. A global G-tension is a function ψ : E → G such that h (W) = 1 for every
ψ
closed walk W.
From now on, a homomorphism between connected directed graphs with a cyclic order
at each of their vertices, is assumed to respect the directions of the edges and the cyclic
orders. Given such a homomorphism s :K → H and any ψ :E(H) → G, the map ψ lifts to
a map ψ : E(K) → G, given by ψ (e) := ψ(s(e)) for every e ∈ E(K).
s s
Lemma 4.1. Let s : K → H be a covering. If ψ is a local G-tension on H, then ψ is a
s
local G-tension on K.
Proof. Let W be a walk around a face F of K. Because s respects the local ordering, we
know thatW maps toaface s(F) of H, wheres(W)mightgo arounds(F)several times, say
f times. Then we have that h (W) = h (s(W)) = h (s(F))f = 1, where the last equality
ψs ψ ψ
follows from the fact that ψ is a local G-tension.
If ψ is a local G-tension on H and s: K → H a covering, then we say that s is a global
covering with respect to ψ if ψ is a global G-tension on K. Let ψ be a local G-tension on H
s
and let T bethe universalcovering graph of H. Fixing a vertex v ∈ V(H), we can bijectively
associate to a vertex of T a non-backtracking walk W in H starting in v. Now label that
vertex with (e(W),h (W)), where e(W) denotes the end vertex of W.
ψ
Let H := T/I, where I is the equivalence relation on the vertices of T given by
ψ
the labeling. We do not allow multiple edges in H , but we do allow loops. If a vertex
ψ
of T has label (e(W),h (W)), its image under the canonical map T → H is denoted by
ψ ψ
[(e(W),h (W))].
ψ
Lemma 4.2. With notation as above, let s :H → H be given by [(e(W),h (W))] → e(W).
ψ ψ
Then s is a global covering with respect to ψ.
Proof. The map s is clearly well-defined and surjective. We show that it is a covering. For
a vertex of H labeled by [(e(W),h (W))], we can extend the walk W in H exactly to
ψ ψ
the neighbors of e(W). If these neighbors are distinct from e(W), then they correspond to
different vertices in H . By our definition of (graph theoretic) neighborhood (see Section
ψ
4.1), this also works in case there is a loop at s(e(W)). This shows that the neighborhood
of v with label [(e(W),h (W))] in H is equal to that of e(W) in H.
ψ ψ
To show that s is a global covering, let W be a closed walk in H starting in [(v,1)]. If
ψ
e(W) = [(e(s(W)),h (s(W))], then we have that h (W) = h (s(W)). Hence for any closed
ψ ψs ψ
walk starting in [(v,1)] it holds that h (W)= 1.
ψs
If W is a closed walk starting in a vertex [(u,h)], with u ∈ V(H) and h ∈ G, different
from [(v,1)], we can conjugate it with a path P from [(u,h)] to [(v,1)]. It is now easy to see
that this walk also has h (W)= 1, as 1= h (PWP−1) = h (P)h (W)h (P)−1.
ψs ψs ψs ψs ψs
9
We use Theorem 3.2 to cellularly embed H in the closed orientable surface S of
g(H)
genus g(H) and write v for the point in S associated with v in H, the vertex that we
g(H)
fixed. Let π (S ,v) be the fundamental group of S based at v. A local G-tension ψ
1 g(H) g(H)
on H gives rise to a group homomorphism
(15) f : π (S ,v) → G,
ψ 1 g(H)
by sending a closed walk W starting in v to h (W).
ψ
Proposition 4.3. It holds that g(H )= k·(g(H)−1)+1, where H is the graph of Lemma
ψ ψ
4.2 and where k is the size of the image of f in (15).
ψ
Proof. Let s be the global covering from Lemma 4.2 and let k be the size of the image of
the homomorphism f from (15). By construction of H = (V(H ),E(H )), we have that
ψ ψ ψ ψ
|s−1(u)| = k for every vertex u of H = (V(H),E(H)). As s is a covering, it holds that
|E(H )| = k·|E(H)|. Moreover, by construction of H and since s respects the cyclic order,
ψ ψ
it is true that |F(H )| = k·|F(H)|, where F(H ) and F(H) denote the collection of faces
ψ ψ
of H and H respectively.
ψ
By equation (10) it follows that
2−2g(H ) = |V(H )|−|E(H )|+|F(H )| = k·(|V(H)|−|E(H)|+|F(H)|) = k·(2−2g(H)).
ψ ψ ψ ψ
By solving for g(H ), the proposition follows.
ψ
We further remark that H can be interpreted as the derived graph of a voltage graph, see
ψ
[4].
Let ψ be a local G-tension on H. A global covering q : K → H with respect to ψ is
called minimal if for every global covering r : K′ → H with respect to ψ, there is a covering
p :K′ → K such that p◦q = r.
Proposition 4.4. Let ψ be a local G-tension on H. The global covering s : H → H from
ψ
Lemma 4.2 is minimal.
Proof. Let W be a walk in K starting in a vertex u, such that t(u) = v. We can label
the end vertex of W in K with (e(t(W)),h (t(W))) (this is well-defined because of Lemma
ψ
4.2). Now let r : K → H be such that it maps a vertex labeled (e(t(W)),h (t(W))) to
ψ ψ
[(e(t(W)),h (t(W)))]. The map r is surjective, because we can lift any walk in H to a walk
ψ
in K with the same height, so any labeling giving a vertex [(e(W),h (W))] in H , can be
ψ ψ
obtained as a labeling in K. Because the labeling is well-defined, we are done.
Let H∗ denote the dual of H in S. Any map ψ defined on the edges of H naturally
induces a map ψ∗ on the edges of H∗. The following lemma is well-known, yet a proof is
given for completeness.
Lemma 4.5. Let ψ be a G-flow on H, then ψ∗ is a local G-tension on H∗.
Proof. Bytheflowcondition,wehavethatifW isawalkaroundafaceinH∗,thenh (W) =
ψ∗
1.
Let s be a covering from K to H. Let ψ be a global G-tension on K, such that
ψ(e ) = ψ(e ), for all e ,e ∈ s−1(e), for every e ∈ E(H). If the map ψ ◦s−1 on edges of
1 2 1 2
H (which is well-defined now) is a local G-tension, then ψ is called a global covering tension
(with respect to s). We call a global covering tension with respect to s minimal if s is a
minimal global covering.
Let D be the map that sends a pair (H,ψ), a directed graph H with a cyclic ordering
at each vertex and a G-flow ψ on this graph, to (H∗ ,ψ∗◦s), where ψ∗ is the dual of ψ and
ψ∗
s :H∗ → H∗ is the covering from Lemma 4.2 of the dual H∗ of H.
ψ∗
10
