scdo 2016 young soo kwon yeungnamuniversity,...

SCDO 2016

Young Soo KwonYeungnam University, Korea

February 18, 2016, Queenstown(Joint Work with Istvan Kovacs))

Outline

1. Introduction to maps, regular maps, Cayley maps and regular Cayley maps

2. Skew-morphisms and their properties

3. Skew-morphisms of dihedral groups

4. Classification of regular Cayley maps on dihedral groups

5. Future research

Introduction to maps, regular maps, Cayley maps and regular Cayley maps

[Definition]1. A = is a 2-cell embedding of a graph

(topologi into a cl

caos

l) maped surface .

G SG S

M



(topologi into a cl

caos

l) maped surface .

G SG S

M

2. For any map = , a is called an of . The set of arcs of is denoted by D(

mutually incident vertex-edge pair G).a

G Src

M

M M



(topologi into a cl

caos

l) maped surface .

G SG S

M

2. For any map = , a is called an of . The set of arcs of is denoted by D(

mutually incident vertex-edge pair G).a

G Src

M

M M

3. Any orientable map = can be des triplecibed by a (D;R,L) such thatG SM

(1) D is the set of arcs of the underlying graph .(2) is a whose orbits coincide with the sets of arcs based at the same vertex.(3) is an

permutation of

involution exchanging tw o arcs iof n

GR

L

D

D cident to the same edge.

R R

L

L

4. A : graph iso. extended to a surface omeo. hmap isomorphism

+ -

5. A : graph auto. extended to a surface homeo.: the set of automorphisms of . ( , resp.) : the set of orientation-preserving(ori

Aut( )A entation-reversing, resp.)

ut ( ) Aut (

)

map automorphismMM

M M

automorphisms of .M


+ -



ut ( ) Aut (

)

map automorphismMM

M M

automorphisms of .M

+6. For any map = , Aut ( acts semi-regularly on D( ). If the action is regular then we call a or a o

f .

regular mapregular embeddi

S G

Gng

G M M)

M


+ -



ut ( ) Aut (

)

map automorphismMM

M M

automorphisms of .M


f .


S G

Gng

G M M)

M

17. If and =( : ) are isomorphic then is called . Otherwise, refl iexible s call chiraled .

G R -1M M M

M-reflexibNote that = is ifle Aut ( )f .G S M M


+ -



ut ( ) Aut (

)

map automorphismMM

M M

automorphisms of .M


f .


S G

Gng

G M M)

M

17. If and =( : ) are isomorphic then is called . Otherwise, refl iexible s call chiraled .

G R -1M M M

M-reflexibNote that = is ifle Aut ( )f .G S M M

[Definition]-1 Cayley graph Cay( : )=(1. For a group and a s ,et such that , a

is a graph suc {{ , } | }h that and . )X V

V E g gX X X

xE

x X

2. For any , let : such that ( ) for any . Let { | }.g g gg L L h gh h L L g

Aut( ( : )) L Cay X


Aut( ( : )) L Cay X

1

3. For a Cayley graph ( : ) and of , is a map ( ; , ) such that

and fo

cyclic permutation

any a Cayley map CM( :X,p)

, ( , ) ( , ( )) ( , ) r and .( , )

G Cay X

D X R g x g p x L g x gx x

p XD R

xL

g

M

Aut ( ( : , )) L CM X p


Aut( ( : )) L Cay X

1


and fo

cyclic permutation


, ( , ) ( , ( )) ( , ) r and .( , )

G Cay X


p XD R

xL

g

M

Aut ( ( : , )) L CM X p

-1 -1

-1 -1 -1

-1 -1

4. For a Cayley map ( : , ) and for any , if ( ) ( ) then ( : , ) is called balanced

anti-balanced

if ( ) ( ) then if ( ) t-balance( ) then . dt

CM X p x Xp x p x CM X pp x p xp x p x


Aut( ( : )) L Cay X

1


and fo

cyclic permutation


, ( , ) ( , ( )) ( , ) r and .( , )

G Cay X


p XD R

xL

g

M

Aut ( ( : , )) L CM X p

-1 -1

-1 -1 -1

-1 -1


anti-balanced



0 11

( )

15. For a Cayley map ( : , ) with ( , , , ),

:[ ] [ x =x ] defined by is called a distribution of inverses .d

i i

G CM X p p x x x

d d


Aut( ( : )) L Cay X

1


and fo

cyclic permutation


, ( , ) ( , ( )) ( , ) r and .( , )

G Cay X


p XD R

xL

g

M

Aut ( ( : , )) L CM X p

-1 -1

-1 -1 -1

-1 -1


anti-balanced



0 11

( )

15. For a Cayley map ( : , ) with ( , , , ),

:[ ] [ x =x ] defined by is called a distribution of inverses .d

i i

G CM X p p x x x

d d

[Lemma]

1

a groupFor a re homo. gular Cayley map ( : , ) ( ; , ), s. t. for any [ ], where (0, 1,

:( , )

,1)

,, -i

R LCM X p D R Li d d

Skew-morphisms and their properties

( )

For a group , a bijection : is called with powerskew-morphism (1 )function : if and for a1 ( ) ( ) ( ) ll , .g g hgh g h


( )


[Lemma]

1A skew-morphism of containing an orbit satisfying and other skew-morph

admiisms

ss: .

iblenonadmi i

:ss ble

O O OO

Classification of regular maps on classification of admissible skew-mor2 p. s hism

regular a skew-mor.1. A Cayley map ( : , ) : s. t. : ( ) |and .XX XCM X p p


( )


[Lemma]


admiisms

ss: .

iblenonadmi i

:ss ble

O O OO



belong to the same r4. g and ight coset( ) ( ofh .e ( )) r Kg h

: a skew-morphism of a group w.r.t a power function 3. . ( ) { | ( ) 1}Ker g g


( )


[Lemma]


admiisms

ss: .

iblenonadmi i

:ss ble

O O OO



belong to the same r4. g and ight coset( ) ( ofh .e ( )) r Kg h 1

0

( ( ))

(gh)= (g)5 ( ) .h.

ji

i

gj j



( )


[Lemma]


admiisms

ss: .

iblenonadmi i

:ss ble

O O OO



belong to the same r4. g and ight coset( ) ( ofh .e ( )) r Kg h

( ) 1( ) 1

0(gh)= (h)+ ( (h)6. )+ + ( (h))= ( ( ) . )

gg i

ih

1

0

( ( ))

(gh)= (g)5 ( ) .h.

ji

i

gj j


1

07. is a skew-morphism is multiple of a module |( | for any ( ) g) .

ji

i

j g j

Skew-morphisms of dihedral groups2 2, | ( ) 1 : dihedral group of order 2n.

Let and A-type element B-ty . pe elemen: , a : t

nn

i in n n

D a b a b ab

A a B D A a b

CM(D : X, p) is all elements in X balanced B-type elemare s.entn

Skew-morphisms of dihedral groups2 2, | ( ) 1 : dihedral group of order 2n.

Let and A-type element B-ty . pe elemen: , a : t

nn

i in n n

D a b a b ab

A a B D A a b

CM(D : X, p) is all elements in X balanced B-type elemare s.entn

[Theorem] (2013, I. Istvan, D. Marusic, M. Muzychuk

2015, Z.Y. Zhang)

1

2

=CM(D : X, p) : regular Cayley map on D s.t. is in Aut( ).

is equivalent to one of tbala

core-free

he following:(1) n=1, CM(D ,{ }, ( )) (2) n=2, CM(D ,{ , ,

ncedABB-typ}, ( , , ))

(3

e

)

nn n AL

b ba b ab a b ab

M M

M

1 2 1 23

1 14

2 2 2 3 2 1

n=3, CM(D ,{ , , , }, ( , , , ))

(4) n=4, CM(D ,{ , , }, ( , , ))

(5) n=2m with odd m, CM

AABB-type

AAB-type

alternati(D , , ( , , , , , , )) n gn nn

a a b a b a a b a b

a a b a a b

a a b a b a a b a a b a

CM(D : X, p): regular H A s.t. CM(D /H: X/H, p): regula cr, ore-freen n n

[Lemma]

=CM(D : X, p) : regular Cayley map on D (1) s.t. (2) The kernel of the corresponding sk

A =dihedral subew-morph groupism is .

n

n n

n

n xX A X A x

M

[Lemma]



n

n n

n

n xX A X A x

M

[Some known results]

2 3

1

1 1(1) balanced with (n,r)=1 and

r + +1 0 (mod n).

, ,

('05, Y. Wang a

, ,

nd R. Q. Feng)

d dr

d

r rp b ab a b a b

2 2 2 12 2 ( 1) 2 ( )

2

1(2) t-balanced with ,

-1(mod n), 2k ( + + )+ -1 0 (mod n) ('06, J. H.

, , , , , , , , : e

Kwak, K and R. Q. Fe AB-tng)

ven

ype

j

j

k k k

j

p b a a b a a b a a b a n

[Lemma]



n

n n

n

n xX A X A x

M


2 3

1


r + +1 0 (mod n).

, ,

('05, Y. Wang a

, ,

nd R. Q. Feng)

d dr

d

r rp b ab a b a b

2 3 4 5 4 2 4 1

(3) n:odd, nonbalanced with

, 1(mod n), j:odd ('13, I. Kovacs, D. Marusic and M. A

, , , , , , ,

Muzychuk) ABB-type

,

3

k k

j

p a a a b a b a a a b a b

n k

2 2 2 12 2 ( 1) 2 ( )

2


-1(mod n), 2k ( + + )+ -1 0 (mod n) ('06, J. H.

, , , , , , , , : e


ven

ype

j

j

k k k

j


[Lemma]



n

n n

n

n xX A X A x

M


2 3

1


r + +1 0 (mod n).

, ,

('05, Y. Wang a

, ,

nd R. Q. Feng)

d dr

d

r rp b ab a b a b

2 3 4 5 4 2 4 1

(3) n:odd, nonbalanced with

, 1(mod n), j:odd ('13, I. Kovacs, D. Marusic and M. A

, , , , , , ,

Muzychuk) ABB-type

,

3

k k

j


n k

2 3 4 5 4 2 4 1

(4) skew-type 3 with

, 1(mod n), j:odd ('15, Z. Y.

, , , ,

Zhang)

,

3

, , ,k k

j


n k

2 2 2 12 2 ( 1) 2 ( )

2


-1(mod n), 2k ( + + )+ -1 0 (mod n) ('06, J. H.

, , , , , , , , : e


ven

ype

j

j

k k k

j


2 1 11 2 2 2(5) reflexible , with AAB-

( , , ,a b,type

.('1

a6(?), I. Kovac

,a )s and K)

4,

8n n n

p b a a n k


( , , ,a b,type

.('1

a6(?), I. Kovac

,a )s and K)

4,

8n n n

p b a a n k

2 3 2 1

2 23 12 2

(6) minimal kernel , with or

with , . ('16(?), I. Kovacs and

( , , , , , , ) : even

( , , , , , AB-type

,K)

) 8n n

p b a a b a a b a n

p b a a b a a b a n k


( , , ,a b,type

.('1

a6(?), I. Kovac

,a )s and K)

4,

8n n n

p b a a n k

2 3 2 1

2 23 12 2

(6) minimal kernel , with or

with , . ('16(?), I. Kovacs and

( , , , , , , ) : even

( , , , , , AB-type

,K)

) 8n n

p b a a b a a b a n


2 53 4

2

3

2balanced ( , , , , , ,(7) skew-type 2 or with , -1 (mod n) for some j. ('16(?), K)ABB-typ

)4e

2

j

n n

t p a a b a b a a b a bn k

Classification of regular Cayley maps on dihedral groups

Any regular Cayley map on dihedral group ( : , ) in the following list. (In fact, all maps

isomorphic in the li

to one ost are regu )

flar

nCM D X p

[Main Theorem]

core-free regular m1. aps.

2 3 1 112. with, , 1 0 (mod n). balan(, ced, )d dr r drp b ab a b a b r r

2 53 4

2 2

3

3.

( , , , , , , )4with , -1 (mod n) for som ABB-te j. y2 pe j

n n

p a a b a b a a b a bn k

2 3 4 5 33 3 12 2 2 24. with( , , , , , , , ) 8 4, -1 (mod ) orn

2

j jn n n nj np b a a a b a a a b n k

42 6 3 3 5 6 3 3 22 2( , , , , , , , ) 8 4, -1

with

(m

AAB-type

od n

)2

j jn n

j np b a a a b a a a b n k

2 2 2 12 2 ( 1) 2 ( 1)6 (2j+1)-balance, , , , , , , ,. (1) with ( ) rd ojk k kp b a a b a a b a a b a

2 3 2 1

2 23 12 2

( , , , , , , ) : even

( , ,

minima

(2) with or

with ( ) ol r , k, , , ) 8 erneln n

p b a a b a a b a n


1 11 1

1 22 3 2 1

1

2 2 2 23 1

1 22 1, coprime odd numbers

(mod 2 ) ( ,

(3) with and are satisfying the follows:

(i) or

, , , , , )

( , , , , , a b, wi) th n n

n n n

p n b a a b a a b a

b a a b a a

n n

2 2 2 12 2 ( 1) 2 ( 1)2

11

1 1

3

(mod 2 ) , , , , , , , ,

gcd(2 , 2 ) divides (ii) namely

minimal kernel

t-b

-1 1 gcd( , ) 1, 2 gcd(2 , ) 1

alanced

.

jk k kp n b a a b a a b a a b a

n j jn j n j

AB-type

2 3 4 5 4 2 4 1

5.

with , 1(mod n), j:odd AABB-ty

, , ,

pe

, , , , ,

3 .

k k

j


n k

[Sketch of proof]

1 (0) (0) 1

10ABB-type ( ,a group homo. : , ,

)

(0) (

3. s. t.

= ) 332 2

nDd di

R L

i

[Sketch of proof]

1 (0) (0) 1


)

(0) (

3. s. t.

= ) 332 2

nDd di

R L

i

2 53 4

0 1

2 2

3

2 3 4

skew-type 2

with , -1 (mod n)

( ( ), ( ), ( ), ( ), ( ), ) ( +1, 1, +1, +1, 1, +1, )2 2 2 2

( , , , , , , for som j.4 e

)2

n

j

n

d d d dx x x x x


[Sketch of proof]

1 (0) (0) 1


)

(0) (

3. s. t.

= ) 332 2

nDd di

R L

i

2 2

2 2 2 1 2

2 22 2 1

2

2 2

( ) , ( ) ,

( ) , ( )

) ,

( i i i i

n ni ii i

a a a a b

a b a

Ker a b

b a b a

wellConv -defersely, ined skesuch w-mor i phs a .ism

2 53 4

0 1

2 2

3

2 3 4

skew-type 2

with , -1 (mod n)

( ( ), ( ), ( ), ( ), ( ), ) ( +1, 1, +1, +1, 1, +1, )2 2 2 2

( , , , , , , for som j.4 e

)2

n

j

n

d d d dx x x x x


2

4. Assume that ( , , ) and the typ

AAB-typeABB

skew-typThe orbit of has e is

the same length with that of

(3 1) 3 (1), (3 2) and

3 (2)

(1)

e

(

4

2 3 )

a ai i

p

i

b a

i

(mod d), 3 (1) 3 ( mod d) 2d

2

4. Assume that ( , , ) and the typ

AAB-typeABB

skew-typThe orbit of has e is

the same length with that of

(3 1) 3 (1), (3 2) and

3 (2)

(1)

e

(

4

2 3 )

a ai i

p

i

b a

i

(mod d), 3 (1) 3 ( mod d) 2d

3

3

4 4 4 3 4 2 1

4 4 2 14 1 42

4

2 2

3

( ) , ( ) ,

( ) , ( ) ,

( ) ,

1

12 5

(

j

j

i i i i

n ni ii i

a a a b a b

a a a b

Ker a a

jb

a

b

a

3

3

4 2 4 2 4 1 4 1

4 3 4 3 4 4 1

) , ( ) ,

( ) , ( ) ,

9 4

3 2

j

j

i i i i

i i i i

a b a j

j

b a

a a b a b a

2 3 4 5 33

42 6 3 3 5 6 3

3 12 2 2 2

3 22 2

( , , , , , , , ) 8 4, with or

with

depending o

-1 (mod n)2

( , , , , , , , ) 8 4, -1 (mo

n or .

5(1)= +1

d

+16 6

n)2

j j

j j

n n n nj

n nj

np b a a a b a a a b n k

nb a a a b a a a b n

d

k

d

2 2 (0) (0) 2 2

0 1 2 3 4

10a group homo. : , ,

AABB-

=

( ( ), ( ), ( ), ( ), ( ), ) (2 (0) 1

type ( , )

(4 ), (0) 1

4 (0),

5. s. t.

(4 1,1, (0) 1,2

) 4 1 an(0) 4 (0) 0d

nR L

x x x x x

D

i i i i

20 1

2

(0) 1, )

( ) (0) 1 ( ), (

and

namely ) ( ) skew-type 3 x x a a

2 3 4 5 4 2 4 1

with , 1(mod n), j:odd

, , , , ,

.

, , ,

3

k k

j


n k

2

2

2 2,

| : group a

6. :

uto.

skew-morph

|

ism of D , furthermore

correspon

A

ds to or

B-t

balanced t-bal

y

anced

pe

aa b

n

2

2

2 2,

| : group a

6. :

uto.

skew-morph

|


correspon

A

ds to or

B-t

balanced t-bal

y

anced

pe

aa b

n

2| balanced corresponds to our map i t-ba as l nceda

2 3 5| like ( ,contains an , minimal kernorbit of length our map i, ) 2

es lan a a a

2

2

2 2,

| : group a

6. :

uto.

skew-morph

|


correspon

A

ds to or

B-t

balanced t-bal

y

anced

pe

aa b

n

11gcd( clo 2 , 2 ) divides -1sed under inverse n j j

2| balanced corresponds to our map i t-ba as l nceda

2 3 5| like ( ,contains an , minimal kernorbit of length our map i, ) 2

es lan a a a

1 11 1

1

2

1 22 3 2 1

1

2 2

2

23

| :

2 1, coprime odd numbers

(mod 2

other

)

t-balanced

with and

( , , , ,

are satisfying

or

, , )

( , , , a , ,

a

n n

n n n

p n b a a b a a b a

b a a b

n

a

n

2 2 2 1

2 1

2 2 ( 1) 2 ( 1)2

b, ) 3

(mod 2 ) , , ,

minimal kernel

t-b, , , , ,

wit

alanced

h jk k k

a

p n b a a b a a b a a b a

Future Research

1. Group structure of each regular Cayley map on dihedral group

2. Classification of regular Cayley maps on dihedral group using group theoretic method.

3. Classification of full (admissible and nonadmissible) skew-morphisms of dihedral group.

scdo 2016 young soo kwon yeungnamuniversity,...

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