The edge-flipping group of a graph

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Contents lists available atScienceDirect

European Journal of Combinatorics

journal homepage:www.elsevier.com/locate/ejc

The edge-flipping group of a graph

I

Hau-wen Huang

1

, Chih-wen Weng

1

Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC

a r t i c l e i n f o Article history:

Received 31 October 2008 Received in revised form 17 March 2009 Accepted 11 June 2009 Available online 5 August 2009

a b s t r a c t

Let X =(V,E)be a finite simple connected graph with n vertices and m edges. A configuration is an assignment of one of the two colors, black or white, to each edge of X . A move applied to a configuration is to select a black edge ∈E and change the colors of all adjacent edges of. Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on X , and it corresponds to a group action. This group is called the edge-flipping group WE(X)of X . This paper

shows that if X has at least three vertices, WE(X)is isomorphic

to a semidirect product of(Z/2Z)k and the symmetric group Sn

of degree n, where k = (n−1)(m−n+1)if n is odd, k = (n−2)(m−n+1)if n is even, and Z is the additive group of integers.

1. Introduction

An ordered pair X

=

(

V

,

E

)

is a finite simple graph if V is a finite set and E is a set of some 2-element subsets of V . The elements of V are called vertices of X and the elements of E are called edges of X . Two vertices u

, v

of X are neighbors if

{

u

, v} ∈

E. A finite simple graph X

=

(

V

,

E

)

is connected if for any two distinct vertices u

, v ∈

V there exists a subset

{{

u0

,

u1

}

, {

u1

,

u2

}

, . . . , {

uk−1

,

uk

}}

of E with u0

=

u and uk

=

v.

Throughout this paper, X

=

(

V

,

E

)

is a finite simple connected graph with

|

V

| =

n and

|

E

| =

m.

This paper focuses on two flipping puzzles defined on the graph X as follows. A configuration of the first puzzle (second puzzle, respectively) is an assignment of one of the two colors, black or white, to each edge of X (vertex of X , respectively). A move applied to a configuration is to select a black edge

∈

E

I _{Research partially supported by the NSC grant 97-2115-M-009-002 of Taiwan, ROC.}

E-mail addresses:poker80.am94g@nctu.edu.tw(H.-w. Huang),weng@math.nctu.edu.tw(C.-w. Weng). 1 Fax: +886 3 5724679.

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(black vertex

v ∈

V , respectively) and change the colors of all edges

0_with

_|

0

_∩

_{| =}

_{1 (all neighbors}

of

v

, respectively). Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping

puzzle (the vertex-flipping puzzle, respectively) on X . A set O of some configurations is an orbit of the

edge-flipping puzzle (the vertex-flipping puzzle, respectively) on X if for any two configurations in

O, one can reach the other by a sequence of moves. The edge-flipping puzzle corresponds to a group

action, and this group is called the edge-flipping group WE

(

X

)

of X

=

(

V

,

E

)

. See Section3for details. The orbits of the edge-flipping puzzle on X have been determined. If X is a tree with at least three vertices, the edge-flipping group WE

(

X

)

of X is isomorphic to the symmetric group Snof degree

n. Wu [18] illustrated both of these results. The main goal of the current study is to produce the complement part of his second result, and show that if X has at least three vertices, then WE

(

X

)

is isomorphic to

(

Z

/

2Z

)

(n−1)(m−n+1)o Sn

,

if n is odd;

(

Z

/

2Z

)

(n−2)(m−n+1)o Sn

,

if n is even,

where Z is the additive group of integers. Section5explains the structure of WE

(

X

)

in detail. The development and history of vertex-flipping puzzles can be found in the literature [4–6,8,11, 9,15,17]. For example, the vertex-flipping puzzles implicitly appear in Chuah and Hu’s papers [5,6] when they study the equivalence classes of Vogan diagrams and extended Vogan diagrams [1,2,14]. Note that the vertex-flipping puzzles are called lit-only

σ

-games in [9,15,17].

The vertex-flipping puzzle on X also corresponds to a group action [15], and this group is called the vertex-flipping group WV

(

X

)

of X . Some properties of the vertex-flipping group WV

(

X

)

of X have been known. For example, WV

(

X

)

has the trivial center, and WV

(

X

)

is a homomorphic image of the Coxeter group W of X . If X is a simply-laced Dynkin diagram, WV

(

X

)

is isomorphic to the quotient group of

W by its center Z

(

W

);

moreover,

|

Z

(

W

)| =

1 or 2. See [10] for details. In other words, WV

(

X

)

can be treated as a combinatorial version of the reflection groups on real vector spaces. Although Humphreys gave a faithful geometric representation of any Coxeter group W in a real vector space [12, Section 5.3], the group structures of W and WV

(

X

)

are worthy of further study.

On the other hand, the edge-flipping puzzle on X is the vertex-flipping puzzle played on the line graph L

(

X

)

of X . The edge-flipping group WE

(

X

)

of X is also the vertex-flipping group of L

(

X

)

. The main result of this study can be used to find the group structures of the vertex-flipping groups of some graphs, which do not need to be line graphs. See Section6for details. Note that line graphs are classified in terms of nine forbidden induced subgraphs [3,16].

2. Edge spaces and bond spaces

Let X

=

(

V

,

E

)

denote a finite simple connected graph with

|

V

| =

n and

|

E

| =

m. In this section we

give some basic definitions and properties about the edge space and the bond space of X . The reader may refer to [7, p.23–p.28] for details. LetEdenote the power set of E. Let F2

= {

0

,

1

}

denote the 2-element field. For F

,

F0

_∈

_E_{, define F}

₊

_F0

_{:= {}

_∈

_E

_|

_∈

_F

_∪

_F0

_{, 6∈}

_F

_∩

_F0

_}

_{; i.e., the symmetric}

difference of F and F0, and define 1

·

F

:=

F and 0

·

F

:= ∅

, the empty set. ThenEforms a vector space over F2and is called the edge space of X . Note that the zero element ofEis

∅

and

−

F

=

F for F

∈

E. Since

{{

} | ∈

E

}

is a basis ofE, we have dim E

=

m

.

In the same way as above, the power setVof

V also forms a vector space over F2with symmetric difference as vector addition, and we callVthe

vertex space of X . Clearly, dimV

=

n

.

For a subset U of V , let E

(

U

)

denote the subset of E consisting of all edges of X that have exactly one element in U. In graph theory, E

(

U

)

is often called an edge cut of X if U is a nonempty and proper subset of V . Let E

(v) :=

E

({v})

for

v ∈

V and notice that E

() =

E

({

x

,

y

}

)

for

= {

x

,

y

} ∈

E

.

Proposition 2.1. Let X

=

(

V

,

E

)

be a finite simple connected graph. Then the following

(

i

), (

ii

)

hold.

(i) Each

= {

x

,

y

} ∈

E lies in exactly two edge cuts E

(

x

)

and E

(

y

)

among E

(v)

for all

v ∈

V

.

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Proof. (i) is immediate from the definition of E

(v)

for

v ∈

V . (ii) is immediate from (i) and the

definition of E

(

U

).

The bond spaceBof X is the subspace ofEspanned by E

(v)

for all

v ∈

V . In the following, we give

some basic properties ofB

.

=

(

V

,

E

)

be a finite simple connected graph with n vertices, and letBbe the bond space of X . Then the following

(

i

)

–

(

iv

)

hold.

(i) B

= {

E

(

U

) |

U

⊆

V

}

.

(ii) dim B

=

n

−

1

.

(iii) For u

∈

V , E

(

u

) = P

_v∈_V_−{_u_}E

(v).

(iv) For u

∈

V , the set

{

E

(v) | v ∈

V

− {

u

}}

is a basis of B

.

Proof. (i) follows immediately fromProposition 2.1(ii). Note that the map from the vertex spaceV

onto the bond spaceBof X , defined by

U

7→

E

(

U

)

for U

∈

V

,

is a linear transformation with kernel

{∅

,

V

}

. Hence dim B

=

n

−

1 and this prove (ii). Let u

∈

V .

Since E

(

V

) = ∅

, we have

E

(

u

) =

E

(

u

) +

E

(

V

)

=

X

v∈V−{u}

E

(v).

This proves (iii). Also, the set

{

E

(v) | v ∈

V

− {

u

}}

is a basis ofBsince it has at most n

−

1 elements and spansBby (iii). This proves (iv).

It is not hard to see that there exists an

(

n

−

1

)

-element subset T of E such that

(

V

,

T

)

is connected since X

=

(

V

,

E

)

is connected. We call such T a spanning tree of E. The following proposition says that

{

F

|

F

⊆

E

−

T

}

is a set of coset representatives ofBinE

.

=

(

V

,

E

)

be a finite simple connected graph with n vertices and m edges. LetE

andBbe the edge space and bond space of X respectively. Then the subset

{

F

|

F

⊆

E

−

T

}

of Eis a set of coset representatives ofBinE, where T is a spanning tree of E

.

Proof. Note that there are 2m−n+1_{cosets of}_B_in_E_{because of dim} _B

₌

_n

₋

_{1 and dim}_E

₌

_{m. It is} clear that

|{

F

|

F

⊆

E

−

T

}| =

2m−n+1_{. For any two distinct F}

_,

_F0

_⊆

_E

₋

_{T , the graph}

₍

_V

_,

_E

₋

₍

_F

₋

_F0

₎₎

_is

still connected since T

⊆

E

−

(

F

−

F0

₎

_{, which implies that F}

₋

_F0_{is not an edge cut of X}

_;

_{i.e., F}

₋

_F0

_6∈

_B_.

Hence

{

F

|

F

⊆

E

−

T

}

is a set of coset representatives ofBinE

.

3. Edge-flipping groups and invariant subsets

In this and in the following section, we rephrase some results of [18] in order to facilitate our work. Let X

=

(

V

,

E

)

|

V

| =

n and

|

E

| =

m. LetEandBdenote the edge space and bond space of X respectively. We regard every configuration of the edge-flipping puzzle on X as an element G ofE, where G consists of all black edges. We give a new interpretation of the moves in the edge-flipping puzzle on X : on each round, we select an edge

∈

E. If

is a black edge, then we change the colors of all edges

0with

|

0

∩

| =

1

;

otherwise, we do nothing. Clearly, the orbits of the edge-flipping puzzle of X under the new moves are unchanged. However, the new move by selecting an edge

of X is corresponding to the map

ρ

:

E

→

Edefined by

ρ

G

=

G

+

E

(),

if

∈

G

;

G

,

otherwise (3.1)

for G

∈

E. From the definition of

ρ

, we see that

ρ

2

is the identity map onE. In particular,

ρ

is

invertible. It is straightforward to check that

ρ

is a linear transformation onE. Hence

ρ

is an element in the general linear group GL

(

E

)

ofE

.

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Definition 3.1. The edge-flipping group WE

(

X

)

of X

=

(

V

,

E

)

is the subgroup of the general linear group GL

(

E

)

ofEgenerated by the set

{

ρ

|

∈

E

}

.

Recall that I is an invariant subset ofEunder WE

(

X

)

if I

⊆

Eand WE

(

X

)

I

⊆

I. In the following, we give some significant invariant subsets ofEunder WE

(

X

).

Proposition 3.2 ([18]). Let X

=

(

V

,

E

)

be a finite simple connected graph. LetEandBbe the edge space and bond space of X respectively. Then each coset of BinEis an invariant subset of Eunder WE

(

X

).

Proof. It suffices to show that

ρ

G is in G

+

Bfor any

∈

E and G

∈

E. By(3.1),

ρ

G is equal to either G

+

E

()

or G. Hence,

ρ

G

∈

G

+

Bsince E

() ∈

B

.

From now to the end of this section, we shall study the group action of WE

(

X

)

on the bond space Bof X . Recall that the set

{

E

(v) | v ∈

V

}

spansB. We first determine the cardinality of the set

{

E

(v) | v ∈

V

}

as follows.

Lemma 3.3. Let X

=

(

V

,

E

)

be a finite simple connected graph with

|

V

| =

n. Then

|{

E

(v) | v ∈

V

}| =

n

,

if n

≥

3

;

1

,

otherwise.

Proof. If X is a one-vertex graph, then

{

E

(v) | v ∈

V

} = {∅}

, and if X is a connected graph with two vertices, say x and y, then E

(

x

) =

E

(

y

)

and hence

|{

E

(v) | v ∈

V

}| =

1. Now suppose n

≥

3. Pick two distinct vertices u

, v ∈

V , we show E

(

u

) 6=

E

(v)

. Since the edge cut E

({

u

, v})

is nonempty and by Proposition 2.1(ii), E

(

u

) +

E

(v) =

E

({

u

, v}) 6= ∅;

i.e., E

(

u

) 6=

E

(v).

Throughout the remainder of this section, we assume n

≥

3 and we denote the symmetric group on

{

E

(v) | v ∈

V

}

of degree n by Sn. Suppose

= {

x

,

y

} ∈

E. FromProposition 2.1(i) and the definition of

ρ

in(3.1), we know that

ρ

fixes all E

(v)

’s except E

(

x

)

and E

(

y

)

. Also fromProposition 2.1(ii), we know that E

() =

E

(

x

) +

E

(

y

)

, and hence

ρ

E

(

x

) =

E

(

y

)

and

ρ

E

(

y

) =

E

(

x

)

. In brief, the mapping of

ρ

on

{

E

(v) | v ∈

V

}

is the transposition

(

E

(

x

),

E

(

y

))

in Sn. Since each element g

∈

WE

(

X

)

is generated by

ρ

for

∈

E, the mapping of g on

{

E

(v) | v ∈

V

}

is like a permutation in Sn. Hence we have the following definition.

Definition 3.4. Let

α :

W_E

(

X

) →

Sndenote the group homomorphism defined by

α(

g

)(

E

(v)) =

gE

(v)

for

v ∈

V and g

∈

WE

(

X

).

Let T be a spanning tree of E, and let WE

(

X

)

Tdenote the subgroup of WE

(

X

)

generated by the set

{

ρ

|

∈

T

}

. We say that X is a tree if E

=

T . The following lemma shows that WE

(

X

)

is isomorphic to Snif X is a tree.

Lemma 3.5 ([18, Theorem 8]). Let X

=

(

V

,

E

)

|

V

| =

n

≥

3. Let

Snbe the symmetric group on

{

E

(v) | v ∈

V

}

of degree n. Then

α(

WE

(

X

)) = α(

WE

(

X

)

T

) =

Sn, where T

is a spanning tree of E. Moreover, WE

(

X

)

is isomorphic to Snif X is a tree.

Proof. Note that

α()

is the transposition

(

E

(

x

),

E

(

y

))

for every

= {

x

,

y

}

∈

E

.

Let A

=

{

(

E

(

x

),

E

(

y

)) ∈

Sn

| {

x

,

y

} ∈

T

}

. Pick any two distinct vertices u

, v ∈

V . Then there exists a subset

{{

u0

,

u1

}

, {

u1

,

u2

}

, . . . , {

uk−1

,

uk

}}

of T with u0

=

u and uk

=

v

. Note that

(

E

(

u

),

E

(v)) = (

E

(

uk−1

),

E

(

uk

)) · · · (

E

(

u1

),

E

(

u2

))(

E

(

u0

),

E

(

u1

))(

E

(

u1

),

E

(

u2

))

· · ·

(

E

(

uk−1

),

E

(

uk

)).

Hence A generates all transpositions in Snand then A generates Sn. Thus, the first assertion holds. For the second assertion, let X be a tree. Since m

=

n

−

1, the edge spaceE is the bond spaceBof X byProposition 2.2(ii). Hence, for g

∈

WE

(

X

)

, if gE

(v) =

E

(v)

for every

v

then g is the identity map onE. This shows that the kernel of

α

is trivial. From this and the first assertion, the second assertion

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Example 3.6. Let X

=

(

V

,

E

)

be the star graph of n

≥

3 vertices. ByLemma 3.5, the edge-flipping group WE

(

X

)

of X is isomorphic to Sn

.

4. Orbits

Let X

=

(

V

,

E

)

|

V

| =

n and

|

E

| =

m, and let T

denote a spanning tree of E. Let WE

(

X

)

denote the edge-flipping group of X . In this section, we give a description of the orbits of the edge-flipping puzzle on X in terms of our language. For this purpose, we fix a vertex u in V for this whole section and choose a nice basis of the bond spaceBof X . Recall that fromProposition 2.2(iii) E

(

u

) = P

_v∈_V_−{_u_}E

(v)

, and fromProposition 2.2(iv)

∆

:= {

E

(v) | v ∈

V

− {

u

}}

is a basis ofB. We call∆the simple basis ofB. For each element G inB, let∆

(

G

)

denote the subset of

∆such that the sum of all elements in∆

(

G

)

is equal to G, and let the simple weight s

w(

G

)

of G be the cardinality of∆

(

G

)

. For example,∆

(

E

(

u

)) = {

E

(v) | v ∈

V

− {

u

}}

and s

w(

E

(

u

)) =

n

−

1

.

Recall that an orbit ofEunder WE

(

X

)

is the set

{

gG

|

g

∈

WE

(

X

)}

for some G

∈

E. Note that the orbits ofEunder WE

(

X

)

are corresponding to the orbits of the edge-flipping puzzle on X and are the minimal nonempty invariant subsets ofEunder WE

(

X

).

Therefore, byPropositions 2.3and3.2, every orbit ofEunder WE

(

X

)

is contained in F

+

Bfor some F

⊆

E

−

T

.

In the following lemma, we give a description of the orbits ofBunder WE

(

X

)

in terms of simple weights.

=

(

V

,

E

)

|

V

| =

n

≥

3 and

let T be a spanning tree of E. Then the orbits of Bunder WE

(

X

)

are the same as the orbits of Bunder W_E

(

X

)

T. More precisely, these orbits are

Ωi

:= {

G

∈

B

|

s

w(

G

) =

i

,

n

−

i

}

for i

=

0

,

1

, . . . ,

n

−

1 2

,

where WE

(

X

)

Tis the subgroup of WE

(

X

)

generated by

{

ρ

|

∈

T

}

.

Sketch of Proof. Recall that fromProposition 2.1(ii) andProposition 2.2(i), the bond spaceBof X consists of E

(

U

) = P

_v∈_UE

(v)

for all U

⊆

V . ByLemma 3.5, both WE

(

X

)

and WE

(

X

)

T act on

{

E

(v) | v ∈

V

}

as the symmetric group on

{

E

(v) | v ∈

V

}

. Hence every orbit ofBunder WE

(

X

)

(or WE

(

X

)

T) is one of

{

E

(

U

) | |

U

| =

i

}

for 0

≤

i

≤

n. Since E

(

u

) = P

_v∈V−{u}E

(v)

, both

{

E

(

U

) | |

U

| =

i

}

and

{

E

(

U

) | |

U

| =

n

−

i

}

are equal toΩifor 0

≤

i

≤ d

n−₂1

e

.

For nonempty F

⊆

E

−

T , the orbits of F

+

Bunder WE

(

X

)

in terms of simple weights is given in the following.

=

(

V

,

E

)

|

V

| =

n

≥

3 and

let T be a spanning tree of E. Let F be a nonempty subset of E

−

T and

∈

F . Then the orbits of F

+

B

under WE

(

X

)

are the same as the orbits of F

+

Bunder WE

(

X

)

T∪{}. More precisely, these orbits are

F

+

B

,

if n is odd; F

+

Be and F

+

Bo

,

if n is even,

(4.1)

where WE

(

X

)

T∪{}is the subgroup of WE

(

X

)

generated by

{

ρ

0

|

0

∈

T

∪ {

}}

,B_e

:= {

G

∈

B

|

s

w(

G

)

is even

}

andBo

:= {

G

∈

B

|

s

w(

G

)

is odd

}

.

Sketch of Proof. We now determine the orbits of F

+

Bunder WE

(

X

)

T∪{_}. Since F

∩

T

= ∅

and by

(3.1),

ρ

0F

=

F for any

0

∈

T . Hence W_E

(

X

)

_TF

=

F , and the orbits of F

+

Bunder W_E

(

X

)

_Tare

F

+

Ω_i for i

=

0

,

1

, . . . ,

n

−

1 2

(4.2) byLemma 4.1. It thus remains to consider the action of additional map

ρ

on F

+

B. Let F

+

G

∈

F

+

B

(6)

Fig. 1. β = θ ◦ α.

E

() + ρ

G

∈

B. We discuss the simple weight of E

() + ρ

G as follows. If u

6∈

then s

w(

E

()) =

2 and s

w(

E

() + ρ

G

) =

(

_i

₊

₂

,

_if

_|

∆

(

_G

) ∩

∆

(

_E

())| =

₀

_;

i

,

if

|

∆

(

G

) ∩

∆

(

E

())| =

1

;

i

−

2

,

otherwise, (4.3)

and if u

∈

then s

w(

E

()) =

n

−

2 and

s

w(

E

() + ρ

G

) =

i

,

if

|

∆

(

G

) ∩

∆

(

E

())| =

i

−

1

;

n

−

i

−

2

,

otherwise. (4.4)

By(4.3)and(4.4), some sets in(4.2)are further put together to become an orbit of F

+

B under

WE

(

X

)

T∪{}and we have the result as described in(4.1). Since

is an arbitrary element of F

,

the orbits

of F

+

Bunder WE

(

X

)

T∪F and under WE

(

X

)

T∪{}are the same, where WE

(

X

)

T∪F is the subgroup of W_E

(

X

)

generated by the set

{

ρ

0

|

0

∈

T

∪

F

}

.

Note that

ρ

0F

=

F for all

0

∈

E

−

(

T

∪

F

).

Hence the orbits of F

+

Bunder WE

(

X

)

and under WE

(

X

)

T∪Fare also the same.

5. More on the edge-flipping groups

Let X

=

(

V

,

E

)

|

V

| =

n

≥

3 and

|

E

| =

m

.

In this section, we investigate the structure of the edge-flipping group WE

(

X

)

of X

.

LetBibe a copy of the bond spaceBof X for 1

≤

i

≤

m

−

n

+

1

.

Recall that their direct sum

Bm−n+1

:=

m−n+1

M

i=1 Bi

is the set of all

(

m

−

n

+

1

)

-tuples

(

Gi

)

im=−1n+1 where Gi

∈

Bi and where the addition is defined componentwise; i.e.,

(

G_i

)

m_i₌−₁n+1

+

(

H_i

)

m_i₌−₁n+1

=

(

G_i

+

H_i

)

_im₌−₁n+1

.

Let Aut

(

Bm−n+1

)

denote the automorphism group ofBm−n+1

_.

β :

WE

(

X

) →

Aut

(

Bm−n+1

)

denote the group homomorphism defined by

β(

g

)(

Gi

)

m −n+1 i=1

=

(

gGi

)

m −n+1 i=1 for g

∈

WE

(

X

)

and

(

Gi

)

m −n+1 i=1

∈

B m−n+1

_.

Recall that fromLemma 3.5the group homomorphism

α

from WE

(

X

)

into the symmetric group

Snon

{

E

(v) | v ∈

V

}

is surjective. The following lemma shows that there exists a unique group homomorphism

θ :

Sn

→

Aut

(

Bm−n+1

)

such that the diagram is commutative (Fig. 1).

Lemma 5.2. There exists a unique group homomorphism

θ :

Sn

→

Aut

(

Bm−n+1

)

such that

β = θ ◦ α.

Moreover,

θ(σ)(

E

(v

i

))

mi=−1n+1

=

(σ (

E

(v

i

)))

mi=−1n+1 (5.1)

(7)

Proof. Since

α

is surjective, if

θ

exists then

θ

is unique. To show the existence of

θ,

it suffices to show the kernel Ker

α

of

α

is contained in the kernel Ker

β

of

β.

If g

∈

Ker

α,

then gE

(v) =

E

(v)

for all

v ∈

V and hence g

∈

Ker

β

since

{

E

(v) | v ∈

V

}

spansB

.

Pick

σ ∈

Snand choose an element h in WE

(

X

)

such that

α(

h

) = σ.

To prove(5.1), it suffices to show that

β(

h

)(

E

(v

i

))

mi=−1n+1

=

(α(

h

)(

E

(v

i

)))

mi=−1n+1

for

v

1

, v

2

, . . . , v

m−n+1

∈

V

,

since

β(

h

) = θ(σ)

and

α(

h

) = σ.

ByDefinitions 3.4and5.1, we obtain that both sides of the above equation are equal to

(

hE

(v

1

),

hE

(v

2

), . . . ,

hE

(v

m−n+1

))

as desired. In view ofLemma 5.2, there is a semidirect product ofBm−n+1and Snwith respect to

θ

[13, p. 155], denoted byBm−n+1

oθSn

;

i.e.,Bm−n+1oθSnis the setBm−n+1

×

Snwith the group operation defined by

((

Gi

)

mi=−1n+1

, σ ) ((

Hi

)

mi=−1n+1

, τ) = ((

Gi

)

mi=−1n+1

+

θ(σ)(

Hi

)

mi=−1n+1

, σ τ)

(5.2) for all

(

Gi

)

m −n+1 i=1

, (

Hi

)

m −n+1 i=1

∈

B m−n+1_and

_{σ , τ ∈}

_S

n

.

Recall that T denotes a spanning tree of E and

|

T

| =

n

−

1

.

Let E

−

T

= {

₁

,

₂

, . . . ,

_m−n+1

}

.

Since

{

i

} +

WE

(

X

){

i

} ⊆

Bfor 1

≤

i

≤

m

−

n

+

1

,

we can define a map from WE

(

X

)

intoBm−n+1oθSnas follows.

γ :

W_E

(

X

) →

Bm−n+1oθSndenote the map defined by

γ (

g

) = (({

i

} +

g

{

i

}

)

im=−1n+1

, α(

g

))

for g

∈

W_E

(

X

).

The following lemma shows that

γ

is a group monomorphism. Lemma 5.4.

γ

is a group monomorphism from WE

(

X

)

intoBm−n+1oθSn

.

Proof. For g

,

h

∈

WE

(

X

),

γ (

g

)γ (

h

) = (({

_i

} +

g

{

_i

}

)

m_i₌−₁n+1

, α(

g

))(({

_i

} +

h

{

_i

}

)

_im₌−₁n+1

, α(

h

))

=

(({

_i

} +

g

{

_i

}

)

_im₌−₁n+1

+

θ(α(

g

))({

i

} +

h

{

i

}

)

im=−1n+1

, α(

g

)α(

h

))

=

(({

_i

} +

g

{

_i

}

)

_im₌−₁n+1

+

β(

g

)({

i

} +

h

{

i

}

)

im=−1n+1

, α(

gh

))

=

(({

_i

} +

g

{

_i

}

)

_im₌−₁n+1

+

(

g

{

_i

} +

gh

{

_i

}

)

_im₌−₁n+1

, α(

gh

))

=

(({

_i

} +

gh

{

_i

}

)

_im₌−₁n+1

, α(

gh

))

=

γ (

gh

).

This shows that

γ

is a group homomorphism. Since each g

∈

Ker

γ

fixes the spanning set

{{

₁

}

, {

₂

}

, . . . , {

_m−n+1

}} ∪ {

E

(v) | v ∈

V

}

of the edge spaceE of X

,

g is the identity map onE

.

Hence Ker

γ

is trivial.

ByLemma 5.4, WE

(

X

)

is isomorphic to the subgroup

γ (

WE

(

X

))

of Bm−n+1oθSn

.

Fortunately,

γ (

WE

(

X

))

is knowable. Recall thatBe

,

defined inLemma 4.2, is an

(

n

−

2

)

-dimensional subspace ofB

.

LetBem−n+1denote the subgroup

m−n+1

M

i=1 Be,i ofBm−n+1

_,

_where_B e,i

:=

Befor 1

≤

i

≤

m

−

n

+

1

.

Theorem 5.5. Let X

=

(

V

,

E

)

be a finite simple connected graph with n

≥

3 vertices and m edges. Then

the edge-flipping group WE

(

X

)

is isomorphic to

Bm−n+1oθSn

,

if n is odd; B_em−n+1oθSn

,

if n is even.

Proof. It suffices to show that for any

σ ∈

Sn

,

there exists g

∈

WE

(

X

)

such that

(8)

and for each 1

≤

i

≤

m

−

n

+

1

,

for any

G

∈

Bi

,

if n is odd;

Be,i if n is even,

there exists h

∈

WE

(

X

)

such that

γ (

h

) = (∅, . . . , ∅,

G

, ∅, . . . , ∅, α(

h

)),

(5.4)

where G is in the ith coordinate. FromLemma 3.5,(5.3)follows by choosing g

∈

W_E

(

X

)

T with

α(

g

) = σ,

since g

{

_j

} = {

_j

}

for each 1

≤

j

≤

m

−

n

+

1

.

FromLemma 4.2,(5.4)follows by choosing h

∈

WE

(

X

)

T∪{i}with h

{

i

} = {

i

} +

G

,

since h

{

j

} = {

j

}

for j

6=

i

.

Since dimB

=

n

−

1 and dim Be

=

n

−

2

,

the additive groups ofBandBeare isomorphic to

(

Z

/

2Z

)

n−1and

(

Z

/

2Z

)

n−2respectively, where Z is the additive group of integers.

Example 5.6. Let X be a cycle of n vertices. Then the edge-flipping group WE

(

X

)

of X is isomorphic to

(

Z

/

2Z

)

n−1o Sn

,

if n is odd;

(

Z

/

2Z

)

n−2o Sn

,

if n is even byTheorem 5.5.

The following corollary says that there is a unique (up to isomorphism) edge-flipping group WE

(

X

)

of all finite simple connected graphs X

=

(

V

,

E

)

with

|

V

| =

n

≥

3 and

|

E

| =

m

.

Corollary 5.7. Let X

=

(

V

,

E

)

and X0

₌

₍

_V0

_,

_E0

₎

_{be two finite simple connected graphs with}

_|

_V

_{| = |}

_V0

_{| =}

n

≥

3 and

|

E

| = |

E0

|

.

Then the edge-flipping group WE

(

X

)

of X is isomorphic to the edge-flipping group W_E0

(

X0

)

of X0

.

Proof. We may assume that V0

₌

_V

_.

_{Recall that the simple basis}_∆_of_B_{is the set}

_{

_E

_{(v) | v ∈}

_V

_{− {}

_u

_}}

for some fixed vertex u

∈

V

.

Define E0

_(v),

_B0

_,

_∆0

_{:= {}

_E0

_{(v) | v ∈}

_V

_−{

_u

_}}

_,

_B0

e

,

S

0

nand

θ

0

correspond-ingly. FromTheorem 5.5, it suffices to show thatBm−n+1

oθSnandBem−n+1oθSnare isomorphic to B0m−n+1 oθ0S0 nandB 0m−n+1 e oθ0S 0 nrespectively. Let

µ :

B

→

B

0_{denote the invertible linear}

trans-formation defined by

µ(

E

(v)) =

E0

(v)

for

v ∈

V

− {

u

}

.

Note that there exists a unique group isomorphism

µ

∗

:

Sn

→

Sn0such that

µ

∗

(σ )(

E0

(v)) = µ(σ (

E

(v)))

for all

σ ∈

Sn and

v ∈

V

.

Hence, there exists a unique bijective map

φ :

Bm−n+1oθSn

→

B0m−n+1oθ0S_n0such that

φ((

Gi

)

mi=−1n+1

, σ ) = ((µ(

Gi

))

mi=−1n+1

, µ

∗

(σ))

for all

(

Gi

)

mi=−1n+1

∈

Bm

−n+1_and

_{σ ∈}

_S

n

.

The map

φ

sendsBem−n+1oθSntoB0em−n+1oθ0S

0

n since

µ(

Be

) =

Be0

.

By(5.1)and(5.2), it is straightforward to verify that

φ

is a group isomorphism, as

de-sired.

6. Applications

Let X

=

(

V

,

E

)

|

E

| =

m

.

In this section, we investigate the vertex-flipping group WV

(

X

)

of X

,

which is a vertex version of the edge-flipping group WE

(

X

)

of

X

.

The vertex-flipping group WV

(

X

)

of X is also a subgroup of the general linear group GL

(

V

)

of the vertex spaceVof X

.

The orbits ofVunder WV

(

X

)

are corresponding to the orbits of the vertex-flipping puzzle on X

.

See [10,15] for details. For

v ∈

V

,

let N

(v)

denote the set consisting of all neighbors of

v.

In the following, we give a formal definition of WV

(

X

).

(9)

Fig. 2. The graph Y .

Definition 6.1. The vertex-flipping group WV

(

X

)

of X

=

(

V

,

E

)

is the subgroup of the general linear group GL

(

V

)

ofVgenerated by

{

s_v

|

v ∈

V

}

,

where s_v

:

_V

→

_Vis defined by

s_vU

=

U

+

N

(v),

if

v ∈

U

;

U

,

otherwise

for U

∈

V

.

The line graph L

(

X

)

of X

=

(

V

,

E

)

is a finite simple connected graph with vertex set E and edge set

{{

,

0

} | |

∩

0

| =

1 for

,

0

∈

E

}

.

From this definition, the edge-flipping group of X and the vertex-flipping group of L

(

X

)

are the same.

For this whole section, let Y

=

(

Z

,

H

)

denote the finite simple connected graph with Z

=

{

0

,

1

,

2

, . . . ,

m

−

1

}

and H

= {{

1

,

2

}

, {

2

,

3

}

, . . . , {

m

−

2

,

m

−

1

}

, {

0

,

i1

}

, {

0

,

i2

}

, . . . , {

0

,

i`

}}

,

where

m

≥

2 and 1

≤

i1

<

i2

< · · · <

i`

≤

m

−

1

.

SeeFig. 2. For example, the simply-laced Dynkin diagrams and extended Dynkin diagrams

e

An

,

e

E7

,

e

E8are such graphs.

Let the value

π

1of Y be











`

X

t=1

(−

1

)

tit

,

if

`

is even; `

X

t=1

(−

1

)

tit

+

m

,

otherwise. (6.1) Note that 1

≤

π

₁

≤

m

−

1

.

Theorem 6.2 ([11, Theorem 3.9]). Let 1

≤

k

≤

m

−

1 be an integer. Then the vertex-flipping group of

Y

=

(

Z

,

H

)

is unique (up to isomorphism) among those graphs Y with

π

1

=

k

.

The aim of this section is to determine the structures of the vertex-flipping groups WZ

(

Y

)

of some graphs Y

.

For this purpose, we define some terms. Two finite simple graphs X

=

(

V

,

E

)

and

X0

₌

₍

_V0

_,

_E0

₎

_{are isomorphic if there exists a bijective map}

_{φ :}

_V

_→

_V0_{such that}

_{

_x

_,

_y

_{} ∈}

_{E if and}

only if

{

φ(

x

), φ(

y

)} ∈

E0for all x

,

y

∈

V

.

We shall denote that two finite simple graphs X and X0are isomorphic by writing X

∼

=

X0

_.

_{A graph}

₍

_U

_,

_F

₎

_{is a subgraph of X}

₌

₍

_V

_,

_E

₎

_{if U}

_⊆

_{V and F}

_⊆

_E

_,

_{and a}

subgraph

(

U

,

F

)

of X

=

(

V

,

E

)

is induced if F

= {{

x

,

y

} ∈

E

|

x

,

y

∈

U

}

.

An (induced) subgraph

(

U

,

F

)

of X

=

(

V

,

E

)

is an (induced) path if U

= {

v

₀

, v

₁

, . . . , v

_k

}

and F

= {{

v

₀

, v

₁

}

, {v

₁

, v

₂

}

, . . . , {v

_k−1

, v

k

}}

for some nonnegative integer k

,

where

v

iare all distinct. The following easy fact will be used later. Lemma 6.3. Let X

=

(

V

,

E

)

|

E

| =

m

.

Let 1

≤

k

≤

m be an integer. Then X contains a path of k edges if and only if the line graph L

(

X

)

of X contains an induced path of k vertices.

We determine the structure of the vertex-flipping group WZ

(

Y

)

of Y

=

(

Z

,

H

)

in some special cases.

Corollary 6.4. Let Y

=

(

Z

,

H

)

be a finite simple connected graph with Z

= {

0

,

1

,

2

, . . . ,

m

−

1

}

and H

= {{

1

,

2

}

, {

2

,

3

}

, . . . , {

m

−

2

,

m

−

1

}

, {

0

,

i1

}

, {

0

,

i2

}

, . . . , {

0

,

i`

}}

,

where 1

≤

i1

<

i2

< · · · <

i`

≤

(10)

Fig. 3. All graphs Y are isomorphic to line graphs.

(i) If Y is isomorphic to a line graph L

(

X

)

for some finite simple connected graph X

,

then

π

1

∈ {

1

,

2

,

m

−

2

,

m

−

1

}

.

(ii) If

π

1

∈ {

1

,

m

−

1

}

,

then the vertex-flipping group WZ

(

Y

)

of Y is isomorphic to the symmetric group

Sm+1of degree m

+

1

.

(iii) If

π

1

∈ {

2

,

m

−

2

}

,

then WZ

(

Y

)

is isomorphic to

(

Z

/

2Z

)

m−1o Sm

,

if m is odd;

(

Z

/

2Z

)

m−2o Sm

,

if m is even,

where Z is the additive group of integers.

Proof. Suppose that Y is isomorphic to L

(

X

)

for some finite simple connected graph X

.

Since Y contains the induced path

({

1

,

2

, . . . ,

m

−

1

}

, {{

1

,

2

}

, {

2

,

3

}

, . . . , {

m

−

2

,

m

−

1

}}

),

X contains a path of m

−

1 edges byLemma 6.3. Note that X has m edges, one more edge besides the path. Hence the left column ofFig. 3completely lists all such graphs X

,

and the right column is the corresponding line graph L

(

X

)

of X

.

By computing the value

π

1of Y

∼

=

L

(

X

)

and using(6.1), we find

π

1

=

1

,

2

,

m

−

2

,

or m

−

1

.

This proves (i). Note that the vertex-flipping group WZ

(

Y

)

of Y is the edge-flipping group of X

,

and its group structure only depends on

π

1byTheorem 6.2. Hence we find WZ

(

Y

)

as listed in (ii), (iii) byTheorem 5.5.

(11)

Fig. 4. An example for the vertex-flipping group of a graph, not a line graph, can be determined.

Fig. 5. All graphs Y withπ1∈ {1,2,m−2,m−1}are not isomorphic to line graphs.

Example 6.5. The graph Y

=

(

Z

,

H

)

inFig. 4is a five-vertex graph containing an induced path of four vertices. By(6.1), its value

π

1is 2. Hence, WZ

(

Y

)

is isomorphic to

(

Z

/

2Z

)

4o S5byCorollary 6.4(iii).

In the case of

π

1

∈ {

1

,

2

,

m

−

2

,

m

−

1

}

,

we use(6.1)to find all such graphs, and there are only two graphs that are not isomorphic to line graphs. We show both of them inFig. 5. Note that their possible values

π

1are in

{

2

,

m

−

2

}

.

ByCorollary 6.4(iii), if Y

=

(

Z

,

H

)

is one of two graphs inFig. 5, the vertex-flipping group WZ

(

Y

)

of Y is isomorphic to

(

Z

/

2Z

)

m−1o Sm

,

if m is odd;

(

Z

/

2Z

)

m−2o Sm

,

if m is even.

Remark 6.6. Theorem 6.2implies that in the class of 2m−1_{graphs Y}

₌

₍

_Z

_,

_H

_),

_{the number of the} vertex-flipping groups WZ

(

Y

)

of Y is at most m

−

1 up to isomorphism. Together withCorollary 5.7, it seems that for a given vertex number, the number of non-isomorphic vertex-flipping groups is not too large, and their classification seems to be visible.

Acknowledgements

The authors thank the anonymous referees for giving many valuable suggestions in the presenta-tion of the paper.

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The edge-flipping group of a graph

European Journal of Combinatorics