graphs of problem labeli - ) , ( game The 1 d L
Jan 01, 2016
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.
?)( 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.
of class theofnumber chromatic game The
. and between isgraph planar 338
of class theofnumber chromatic game The
. and between isgraph planar 178
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)
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
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
vertices theselabel toused
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
vertices theselabel toused
becan },,,,{in numbers theall 43210
vertices theselabel toused
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
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}).