game

graphs of problem

labeling-),( game The 1dL

game

chromatic

number

colors. of set a and graph aGiven XG

on played gameperson - twoheConsider t

moveA turns.alternate Bob and Alice

vertexuncoloredan selecting of consisting

XGv fromcolor ait assigning and of

previously assignedcolor thefrom distince

vertices theall if winsAlice . of neighbors to v

:G

wins.Bob else labeled,ly successful are

G ofnumber chromatic game

for which set color a ofy cardinalitleast X

:)(Gg1 game chromatic number of G in

strategy winninga has Alice

the game Alice plays first

the game Bob plays first

:)(Gg2 game chromatic number of G in

Question.

?)( 41 Cg

Alice Bob

241 )(Cg

2

have also we,)()( Since 1 GGig

1

341 )(Cg

341 )(Cg

Alice ?

Question.

?)( 42 Cg

Alice Bob

242 )(Cg

1

have also we),()( Since GGig

1

242 )(Cg

242 )(Cg

Alice 2 Bob2

Theorem. (Faigle et al.)

(Kierstead and Trotter)

Theorem. (Zhu)

of class theofnumber chromatic game The

. is forests 4

Theorem.


. and between isgraph planar 338


. and between isgraph planar 178

Theorem. (Zhu)


. and between isgraph r outerplana 76

L(p,q)-labeling

GqpL of labeling-),(

enonnegativ to)( from function a GVf

satisfying integers

1 ),( if |)()(| (1) vudpvfuf

kan greater th

2 ),( if |)()(| (2) vudqvfuf

GqpLk of labeling-),(-

is label nosuch that labeling-),(an qpL

GqpL ofnumber labeling-),(

:)(, Gqp Gk such that ,number smallest

labeling-),(- a has qpLk

))( replace to)( use we,(when , GGq qpp 1

Question.

?)( 4C

0

4

3

1

44 )(C 4123 CLf of labeling-),(- a is

2by labeled is vertex one

2

3210 ,,,by labeled are cesfour verti the

44 )(C

Since ?

0

44 )(C

Theorem. (Griggs and Yeh)

complete- is problem labeling-),( The NPL 12

graphs. generalfor

Conjecture. (Griggs and Yeh)

).()( ,)( with graph any For GGGG 22

.)()()( ,graph any For 22 GGGG

Theorem. (Gonçalves)

L(p,q)-labelinggame

integers enonnegativ and graph aGiven G

:on played game G

person- twoheConsider t . ,,, 0 qpqpk

moveA turns.alternate Bob and Alice

vertexunlabeledan selecting of consisting

in number ait with label and of aGv

thatconditions thesatisfying },...,,,{ k210

then,),( and by labeled is If (1) 1vudbu

.p|b-a|

then,),( and by labeled is If (2) 2vudbu

.q|b-a|

lysuccessful are vertices theall if winsAlice

wins.Bob else labeled,

GqpL ofnumber labeling-),( game

strategy winninga has Alice that so ,smallest k

:)(,~

Gqp

1

:)(,~

Gqp

2

Alice plays first

Bob plays first

Question.

?)(~

4

2

1 C

Alice

Bob Bob

44

2

1 )(~

C

2a

2

2a

+3

54

2

1 )(~

C

0

Alice 61 or a

using game thiscompletecan Alice},...,,,{in numbers the 6210

64

2

1 )(~

C

64

2

1 )(~

C

Question.

?)(~

4

2

2 C

Alice

Bob

44

2

2 )(~

C

2

1ba

using game thiscompletecan Alice},...,,,{in numbers the 5210

54

2

2 )(~

C 54

2

2 )(~

C

Note.

general!in holdnot dose )()(~~

GG2

2

2

1

Lemma. ,graph any For G

. and ,for 221 di

),( )()(~

21 iGG d

d

i

Lemma. maximum graph with a is If G

)()(~

22 dGd

i

then, degree

Theorem. 22 nKn )(

0 2

4 6

Example.

Alice 0 Bob2

22

2

1 )(~

K

Alice plays firstBob plays first

13

32

2

2 )(~

K

Example.

Alice plays first

Alice 2

Bob0Alice 4

43

2

1 )(~

K 53

2

2 )(~

K

Bob plays first

1

3

Bob 5

Alice

Bob

2

5

Alice0

Bob7

74

2

1 )(~

K

Example.

Alice plays first

Alice

Bob

5

Alice

Bob?

64

2

1 )(~

K

3a

3b

74

2

1 )(~

K

Bob plays first

Bob 1

Alice 5

43

2

1 )(~

K

733

2

14

2

2 )()(~~

KK

Bob plays first

Bob

Alice

74

2

2 )(~

K

74

2

2 )(~

K

4b

2

Bob

Alice

2or 0 ba

Question.

?)(~

nK2

1

?)(~

nK2

2

Observation 1.

412

2

2

2

1 )()(~~

KKK nn Alice

2

0 21 3 54 6 …7

1

Bob

5

Bob

0

Observation 2.

31

2

1

2

2 )()(~~

nn KK

Alice

2

0 21 3 54 6 …7

Bob

1

5

Theorem.

3

972

1

nKn )(

~

3

772

2

nKn )(

~

0 21 3 54 6

3 vertices, 7 numbers

37n

Question.

How to prove this theorem?

(Use induction, we already know that

,)()(~~

412

2

2

2

1 KKK nn

))()(~~

31

2

1

2

2 nn KK

Idea. Alice’s strategy

},,{in numbers theall that so , 21 iiii

with Labels . label toused becan that vv

. with Labels . label toused becan 2ivv

ji number a choose exists, such no If

smallest a choose unlabeled, still is If v

.j

Idea. Bob’s strategy

can },{in numbers theall that so , 1iii

with Labels . label toused becan that vv

. with Labels . label toused be 1ivv

ji number a choose exists, such no If

smallest a choose unlabeled, still is If v

.j

Example.

0 21 3 54 6 7 98 10 1211

13 1514 16 17 1918 20 21 22

2nd

…

4th

1st

6th 3rd

5th 7th

)(~

14

2

1 K

)(~

111

2

1 KK

0 21 3 4 5

6 87 9 10 1211 13 14 15

2nd

…

1st

4th

3rd 5th

Theorem. , , allFor 01 rn

and

dnn

d

rKKn

d

]mod)[()(

)( ~

1

313

314

1

dndn

d

rKKn

d

]mod)[()(

)( ~

2

3223

114

1

Theorem. , ,,, allFor 1nmnmd

2 nmdK nmd )( ,Example

.

211111575711 )()( ,K

0 21 3 54 6

1817 19 2120

?)( , 5711 K

Question.

?)( ,

~

57

11

1 K

:guess

Alice

0 17

Bob

11571157

11

1 )()( ,,

~

KK

},...,{},...,{in numbers theuset can' 2718167

Y

X

Bob’s strategy

proof. 3157

11

1 )( ,

~

K

15a

27Alice Bob

vertices theselabel

toused becan numbers than less 7

16a

4

Bob

211016

Alice

14a 17a

15

vertices theselabel

toused becan numbers than less 53157

11

1 )( ,

~

K

Y

X

proof. 3257

11

1 )( ,

~

KAlice’s strategy

Alice

16

Bob

Alice

22b 10b

28

vertices theselabel toused

becan },...,,{in numbers theall 322928

4


becan },,,,{in numbers theall 43210

15

Alice

17

Alice

)

or ,(

2118

1411

b

b

2118711

1411711

bb

bbc

if,

, if,Y

X

Alice 30

vertices theselabel

toused becan , numbers the 10

12

29

Bob

11

Alice

theusing game thiscompletecan Alice

a labels Bob if },...,,,{in numbers 32210

step second at the in vertex X

)( 54 bAlice

22

Bob

)( 2827 b

10

proof. 3257

11

1 )( ,

~

KAlice’s strategy

Y

X


becan },,,,{in numbers theall 43210


becan },...,,{in numbers theall 322928

16

)9or ,( 32230 bb

.9 if,

, if,

322711

30711

bb

bbc

19

1

vertices theselabel

toused becan , numbers the 10

Bob

2

20

Alice

theusing game thiscompletecan Alice

a labels Bob if },...,,,{in numbers 32210

step second at the in vertex Y3257

11

1 )( ,

~

K

Theorem. , ,,, allFor 2 nmdnmd

)]mod)[( (where 2nmm,n

nmnm

d

nmdK ,,

~

1 )( 22

Thanks!Thanks!

Note.

It is not true that if Alice(resp. Bob) plays

first, and at some step, he can move twice,

then the smallest number needed to complete

the game is less than or equal to )(~

G2

1

)(~

G2

2

(resp.

). For example, ,)(~

52 14

2

1 KC

but if at the first step, Alice need to move

twice, then the smallest number needed to

complete this game is 6({0,1,2,…,6}).

game

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