HAL Id: inria-00073353
https://hal.inria.fr/inria-00073353
Submitted on 24 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Circular-arc Graph Coloring and Unrolling
Christine Eisenbeis, Sylvain Lelait, Bruno P Marmol
To cite this version:
Christine Eisenbeis, Sylvain Lelait, Bruno P Marmol. Circular-arc Graph Coloring and Unrolling.
RR-3336, INRIA. 1998. �inria-00073353�
a p p o r t
d e r e c h e r c h e
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Circular-arc Graph Coloring and Unrolling
Christine Eisenbeis , Sylvain Lelait , Bruno Marmol
N ˚ 3336
Janvier 1998
TH ` EME 1
#%$&('*)"+('-,.0/1'2)".3,546.7'2)(8:9<;=5>-?A@B'*,DCE.3>F@B'*+HGI9JK&(L,MONP@&"QOM>FR
SETVU
W"XWY[Z\^]W"_`W"a3bdceWf_`gh_if
U
W"XW(_
jlk`mon
W"f6prq
\[a7sBs mtk
fEuhW
k
W"v T W kv T
Wxwy7q3q7q7z:Z|{3a3w3}d~W
k
Y"37ZY"zsa7 7W"_
"H77 SET
W k W 7~_`fiW
k
sBW k ~m
uh~va7
m
va7fi~
m
wes k`m3
W(X~_6}d~W"lW(ua3_lbw kim
~w a3wuv m
m3k
~wh %f T W
bBwuhW
k
gt~wh _if
k
bvf`b k W
mo
v~
k
vbha ki
a k v
k
a7s TVSET
Ws k`m3
W(X~_f m:
wu
k
W"a7fi~
m
w_
WfF6WW"wf T W
bBw kim
~wh uhW
k
W"Wa7wuf
T
W[v Tk`m
Xa7fi~vEwtbBX
W k"m3k
f T
~_lsb k s m
_`WElWruh~_`fi~wh tbh~_
T
vghv~vv m
mtk
~wh t_
f T aof<v"a7w
W
m
bwBu
gXW"a7wB_
mo
f T
WK ¡¢¡£-¤¦¥d§§7¨`©«ªh¬1a3wuIw m w
vgdv~v
m
whW(_
m3k
T
~v
T
6Ws k`m
}7Wf T W
a3_ighXshf
m
fi~vs kim
sBW
k
fFg¯®W"fl°
W1f
T
WE~uhf Temo
f T W m3k
~ 7~waoA~wtf`W
k
}±a7
a7X~g
SET
W(wf
T
W²
bw k`m
W(u
k a3s
T
~_E°
m3k
°³´Y
v m m3k
a
W
mtk
²a
k
3WrW(w
m
bh TV
µ¶t·¯¸º¹K» H¼7 k
W 3~_`fiW
k
ao
m
v"aof`~
m
wV½B
mdm
sbw k`m
~wh ½Avghv~vv m
m3k
~w B½a3vghv~vv m
mtk
~wh B½Av~
k
v"bha ki
a k vr
k
a7s
T
¾¦¿À
Á`Â*Ã(Ä
À
ÁrÅÆÂ2Ç2ÈÉ
ÊV˱ÌÎÍdÏVЫÑ*ÒÓÏ5Ô¢ÍÕHÔ¢Ñ-Ö-Ì×Ô¢ØÎØÙÍ*Ú(Õ±ÕoЫÑ-Ö-ÛiÜÞÝÙÞÔßtÌÎÍ-Ûáà¦â1ÛiÌÖ-ã±ÛáÑBäÖ-ÌÎÕoÛiã(ܱÌÎÚ±å0æÑ-ЫåÖ*˱ÛVçÓÚ±Í-Ö-Ñ*Ì×Ô¢ãlä"èáÌÎÛáã±èáÛ5é±Ú±ã±Üê¦é±Ð«ã±Ü±Í
ë
Ú(ÑÓéVìЫÑ-ܱÛáÑ*Ú±ã±í6ܱÛáÑÏÌÎÍ*Í-Ûáã±Í*è2˱ԺæÖ-ØÎÌÎè2˱Ûáãîé±Ð«Ñ-Í*è2ËÚ±ã±íï2ð
ñ6ò Ë(Ñ*ÌÎÍ-Ö-ÌÎã±Û«ðóÌÎÍ-Ûáã"ÝoÛáÌÎÍFôÌÎã±Ñ-Ì×Ô(ðæÑ
õ<ö
ã(Í-Ö-ÌÖ*Ú(ÖÓæFìÚ(Ñ ò ЫåEÕ±Ú(Ö-ÛáÑ*Í-Õ±Ñ2Ô¢è2Ë(Ûiã3÷7ÊdÛiè*˱ã±ÌÎÍ*è*Ë±Ûøã±ÌùÛáÑ*Í*ÌÖ`ìÔºÖú^ÌÎÛáãt÷oçÓÑ*í«ÛáãÖ-ÌÎã±ÌÎÛáÑ*ÍûÖ*Ñ*Ô¢ü«ÛÞý(÷HçÓà2þiÿÿ6ú ÌÎÛáãt÷oçÚ(Í-à
Ö*Ñ-Ì×Ô(ðAóà¦åÔ¢ÌÎØBÍ-Ù"Øù«Ô¢ÌÎãHôèiЫåEÕ(Ø×Ô¢ã±í±ðÖ*ÚÏÌÎÛáãtðÔ¢è«ðÔºÖ
ö5ö
ç
Ë Ð«ã±ÛFà¦çÓØÎÕoÛiÍi÷%çùÛiãÚ±ÛîÜ±ÛØóÚ±Ñ-ЫÕ7Û«÷
ö
ä"Ê÷¯édà«ý«ÿâ1ЫãÖ*ÝoЫã±ã±Ð¢Ö1ä(Ô¢ÌÎãÖ[âîÔ¢Ñ-Ö-ÌÎãt÷5é±Ñ*Ô¢ã±èáÛð
óBà¦åÔ¢ÌÎØ¯Ñ-Ú±ã±Ð±ðâîÔ¢Ñ*åEЫØôÌÎã(Ñ*Ì×Ô(ðæÑ
Unit´e de recherche INRIA Rocquencourt
" "
¶ ¶
Ws kimt
U
W"XWKuhWa7
m
vaof`~
m
ws6]W k~m
uh~dbhWuhW
k
W" 7~_if
k
W(_W"_`f[}hbhWv
m
XX%WW%v m
mtk
~ao 3W
W"fIW:ul]W k`m
ba7 7W uWa_`f
k
bvfºb
k
W0_
m
b_
n
a7vW"wtfiWdbW"_`fIW
k
a3s
T
Wu~wtf`W
k
}±a7W"_v~
k
v"bha7~
k
W"_
W
s
kimt
U
W"XWW"_`fuhWÞf kim
bh}7W
k
uhW"_
k
W"a7fi~
m
w_5W"wtf
k
W<WÞuhW
k ]WuWlu6]W kim
bhao 7WWfW6w m X
k
W<v TBkim
Xa7fi~dbhW
U
p
vWf5W! AW"fÓw
m
b_5uh~_`fi~wh 3b
m
w_5W"_Óv m
mtk
~ao 3W"_¯vghv~tbW"_"½"dbh~tsBW(bh}7W(w3f$#Wf
k
Wf kim
bh}Þ]W"_
U
aEao~uhWÞub I¡¢¡£*¤¦¥t§
§7¨`©«ªh¬d½1uhW(_ v m m3k
~a7 7W"_a7vghv~dbhW"_"½[s m b k
W(_%dbhW"_^w
m
b_ s kim
bh}
m
w_eas k`m
s k
~]Wfh]Wa7_`ghXsf
m
f`~dbhW
_ºbh~}Ha7wtfiW®K_
m
~fî°^&á]W"sBao~_`_`W"b
k
XaHch~XaoWeuhWIa
a3X~Weu'~wtfiW
k
}HaoW(_½ao
m3k
_1WI
k
a3s
T
WIu6]W kim
bh]WK²
m
~_W(_ifv m
m3k
~a
Wa±}3W"vr°
m
b^°³´Yîv
m
bhW"b
k
_s m b k
²0a3_`_`W!(
k
a7wu
)x» ( ¸ +*¶
a7
m
va7fi~
m w uhW
k
W 7~_`f
k
W"_"½[u6]W kim
bha7 7W uW
m
bvW3½v m
m3k
~a7 7W vghv~dbhW3½rv m
mtk
~ao 3W
a3vghv~tbW7½h
k
a7s
T
Wu'~wtfiW
k
}HaoW(_Ev~
k
v"bhao~
k
W(_
¤¦¨©o¨i©H¨ ¨`©¢ª¬ ¨«¤¦¥d§^©o¥Ó¥B¨ פ¦¥d§ q
0
~k
vbha ki
a k v
k
a3s
T
_^a
k
WaO_ºb
va3_`_
mo
~wtf`W
k
_iW(vfi~
m w k
a7s T _
p[wPbwu~
k
W"vfiW"u k a7s
T
~_^a
v~
k
vba ká
a k v
k
a7s
T ~ f T W k
WWch~_`f`_a
a3X~g mo
~wtfiW
k
}Hao_
m w av~
k
vW T
~v
T
Xa7s_%f
T
W^}7W ká
f`~vW"_
m7
½a7wu:fF
m
}3W
k
fi~vW"_1a
k
WIa7u
n
a7vW(wtf~
a7wu
m
whg0~
f T
Wev mtk`k
W"_ºs
m
wuh~wh ~wtfiW
k
}Ha7_
T
a±}3Wa
w m
whW(XsfFg~w3f`W
k
_iW(vf`~
m w ~k
vba ká
a k v[
k
a3s
T
_a
k
Wb_`W"u^~w^_`W}3W
k
aoVa7sBsh~va7fi~
m
w_
ÞSET
W"~
k v m m3k
~wh
T
a7_
W"W"w s k
~Xa
k
~g_ifºbuh~W"u
g S
bvoW k
Y
1
~wu~wh ^f
T
Wv Tk`m
Xa7fi~vKwdbX
W krmo
f T W(_iW%
k
a7s T _
T
a7_
WW(wIs k`m
}7W"wf m%
W[a7w j<
v m
XsW"fiWs k`m3
W"X
g !a
k
W"g%W"fÞao
"
$#
wIf
T
W[sa
k
fi~vba
k
va3_iW m7
s
kim
sBW
k
v~
k
vbha ki
a k v
k
a3s
T
_"½f
T
~_s kimt
W"X
T
a7_rs
m
ghw
m
X~a7Óv
m
XshWch~fFg
Y(qh½Y&%&
('
f T W k[k
W"_ºbhf`_
v m wvW
k
wh~wh v~
k
vba ká
a k v
k
a3s
T
_va3w
W
m
bwBu ~w
zB½*)d½B
' b k s k
~Xa
k
g~wtfiW
k
W(_if~wOv~
k
vba ká
a k vI
k
a7s T v m
mtk
~wh 6a3_X
m
fi~}HaofiW(u
g f T
Ws k`m3
W"X m7
k
W 3~_`fiW
k
ao
m
v"aofi~
m
w^~w
mdm
s_EW"wBv
m
bBw3f`W
k
W"u ~ws kim
k a7X_
m3kT
~
T
sW kim3k
Xa7wvWX~v kim
s
kim
vW"_º_
mtk
_
# w m
shfi~X~("~wh ev
m
Xsh~W
k
_"½
k
W 7~_`fiW
k
a7
m
vaof`~
m
w~_Ef
k
a3uh~f`~
m
waogsBW kámtk
XW(u
g v m m3k
~w +-~
W
a7
mo
v"aof`~wh m w k
W" 7~_if`W
k
_-,f
T
W¤¦¥B£F¡¨/.¡¨`¡¥¢¡[§7¨`©«ªh¬
$0
W k fi~vW(_
m71
a k Wrf
T
W1~
W"fi~XW"_
mo
f T Wîs
k`m
k a7X
}Ha
k
~a
W"_
Y2
SET
W k
W^~_Ia3w W"u 7W
W"fFlW"W"w fF
m
}7W
k
fi~vW(_-3H}Ha
k
~a
W~
Wf`~XW"_~
a3wu
m
whg ~ f T W
}Ha
k
~a
W"_a
k
W_i~XKbhf`a3whW
m
b_`g a7~}7Waof1_
m
XWIs
m
~wtf mo
fT WIs
k`m
k a3X
Y2
#
w f T Weva3_iW
mo
_i~XshW
mdm
s_"½df
T
W~w3f`W ki
W k W"wvW[
k
a3s
T
~_6av~
k
vbha ki
a k v
k
a3s T4"
½5)6
p[w a7uuh~fi~
m
wBao
W"a7f`b kW
m7m
b k s k`mo
W(X ~_1f
T
a7f
m
shf`~Xaol_`v
T
W(ubhW mo
~w_`f
k
bvfi~
m
w_îXa±g v k W(aofiW}Ha
k
~a
W%~
Wfi~XW"_%_`sa3wwh~wh X m3k
W
f T a7w
m
whW~f`W
k
aofi~
m w f T W k W
mtk
W m }7W
k
a7sBsh~w [E~f T f T
W"X_iW"}3W"_
SET
~_XW"a3w_f
T
aoffF
m v m
wB_iW(vbhf`~}3W
~w_`f`a3wvW(_
m7
f T W1}Ha
k
~a
Wrv"a7wew m f
W7oW(shfl~wef
T
W1_`a3X%W
k
W 7~_`fiW k"
p s m
_º_i~
W T a k
u6a
k
W[_
m
bhf`~
m w
~_rf m T
a±}7WKa ¨
£F©H£*¤¦¥t§¨º¡*§7¤98«£2¡¨;:<¡
%=*½¯
T W k
W%f T W k W W k
W"wvWKs
m
~w3f`W
k
~_1_
T ~
fiW"u
m
whWvghvW%a
T
W"a3u
a7fEW"a3v
T
~fiW
k
a7fi~
m w mo
f T WK
mdm
s5½~fv"a7w
WK_iW"W"wa7_Ea
k
~wh mok
W" 7~_if`W k _
<SET
WX%~wh~XaoÓ_`~(WwhW"W"uhW(u
mtk
a
k`m
fºaof`~wh
k
W 3~_`fiW k
WI~_îv
m
XsbfiW"u:~w>dW"vf`~
m
w?%
p_
mo
fFa
k
W_
m
bhf`~
m w ~_1f
m
bwtE~wuf T W
mdm
s:a _ºbh~f`a
WIwdbX
W k ² mo
fi~XW"_
m3k
uhW(_`v k ~
~wh Wcsh~v~fig0f
T
WIao
m
va7fi~
m w ~wtf
m
uh~ AW
k
W(wtf k W
3~_`fiW k _
SET
W k
W"_ºbhfi~wh ~wtfiW ká
W k W(wvW
k
a3s T@BA
+*va7W"u²1Ch¥B¨
¡-§o¨`©«ªh¬D,[v"a7wf
T
W"w
W%v m
m3k
W"u
prvfºbaog3½h
k
a7s
T
bw k`m
~wh
T
a3_
m f T W k
W ¯W"vf`_f
T
a7w
n
b_`f
k
W(wuhW
k
~wh f
T
W1
k
a7s T v m
m3k
a
W
1#
wuhW"W"uV½
f T Wv
Tkim
Xaofi~vewdbX
W kFE
+
BA
,
m7
f T W²
bw kim
W"u
k
a3s
T
va3w}Ha
k
gu
k
a3Xaofi~va7gE~f
T ² a7_%~_
_
Thm
w ~w
~ tb
k
W(
HG
wtE~wuh~wh If
T
Wî
mdm
s^u
m
W"_Ew
m
fuh~_ifºb k`
f T W_`v
T
W(ubhW mo
~w_`f
k
bvfi~
m
w_
bf~f
~wv
k
W(a7_`W"_Ef
T
W_`~(W m7
f T Wv
m
uhW
T
~v
T
~_a7w m fT W k
~Xs m3k
f`a3wtf
a7vf mtkmo
sW kim3k
Xa7wvW~w0X
m
uW k w
X~v k`m
s
k`m
vW(_`_
m3k
_
ÓSET
W kW
mtk
W~f6~_6~Xs m3k
f`a3wtfÞf m
Wa
Wf m v m
w3f kim
f
T
W1bw k`m
~wh KuhW"
k
WWa7_llW"
a3_f
T
WwdbX
W kmok
W" 7~_if`W
k
_lwWW"uW"u_i~XKbfºa7whW
m
b_`g
#
wf
T
~_sBa7sW
k
½6W
Ak
_`fîs
k
W(_iW(wtf m b
kûk
a7XW
m3k
A½5vaoW"uf
T
W I¡¢¡£*¤¦¥t§ §o¨º©¢ªh¬
I*½
m3k
wu~wh
aJKIפLv m
m3k
~wh m7
f T W a7X~g
mo
~wtf`W
k
}±a7_
~k
v"bha ki
a k vK
k
a7s T v m
mtk
~wh ^~_}t~W6W"u a3_îauhW
v m Xs
m
_`~f`~
m w mo
f T
WXWWf`~w ^
k
a3s
T
~wtf
m
v~
k
v"bh~fº_
SET
W"w: 7W"wW
k
aov m
m3k
~wh M+-vghv~va7wu:a7vgdv~v&,
~_îv
m
w_`~uhW
k
W(u a3wu 6Ws kim
}3W%f T W
m
m
E~wh
k
W(_`bf"®
Wfî°NX~w*O&PIQRTS$+
E + O
,U,½+-°
T
a3ssBW(w_1f
m
Wf
T
WWV¤LH£¬
mo
f T W1~w3f`W
k
}Hao
a7X~g5,Þf T W k
WWch~_ifº_6a
m
bwuYXDZ
ûk`m
X
T
~v
T
W}7W kg O
½[X\]XIZ7½h~_
°³ Y v m m3k
a
W
pr_îf
T
Wew
m
uW"_
mo
f T WI
k
a3s
T
Xa7f`v T f T
WI~wtfiW
k
}Hao_½lWIb_`Wf
T
WIf`W
k
X_1~wtfiW
k
}Haov m
m3k
~wh a7wBu
v~
k
vba ká
a k v
k
a7s T v m
m3k
~w ~wtfiW k v T
a7wh 3W"a
g
M#
w f T W tb
k
W"_"½f
T
W^v~
k
vW(_%a k W k
W"s
k
W(_iW(w3f`W"u
g
~whW"_E
T W k
Wîf
T
Wa7_`fs
m
~w3f[~_[f
T
W_ºa7XWa7_f T W
Ak
_`f
m
wW7½f T W k W
mtk
Wî~wtfiW
k
}Hao_E_`sBa7wwh~wh If T W
k
_if
s m
~wtfa
k
Wv"bhf
¬d¨«¤98«£*¤¦¥A¡¤98¡¥º¡¤98*K ±©o¤¦¥¡2©H¤¦£¨h¥ ©H¨«
" E
'
w%a[v~
k
vWlE~f T
s m
~wtf`_a
WW"u
ûkim
X[f m
Y7½±6Wlv
m
w_`~uW k a
a3X%~g7 mo
~wtfiW
k
}Hao_ÓuW~X~fiW"u
gefF
m s m
~wtfº_
ÞSET
Ws kimt
W"X ~_Ef
m
a3_`_`~ 3waIv m
m3k
+,f
m
W"}7W
k
ge~wtfiW
k
}Hao_ºbv T f T
aofEfF
m
m
}3W
k
a3ssh~wh I~w3f`W
k
}Hao_ra
k
Wa7_º_i~ 3whW(uuh~ AW
k
W(w3f1v m
mtk
_ p _ºsBW(v~
v~fFg moÞm
b k s kimt
W"X ~_f
T
aoff T W
W"w 7f T:mo
f T W~wtfiW
k
}Hao_1Xa±g
W
k
W"a7fiW k f T
a7w
m
whWIv~
k
vbX
W k
W(wvW mo
f T Wev~
k
vW7½V~7W~wtfiW
k
}Ha76Y
~w^f
T
Wî~wtfiW
k
}Hao
a3X~g m7
~ 3b
k
WY
p[wtg ~wtfiW
k
}Ha7e~_
T
W"wvWuhW
wW"u
g~f`__`f`a
k
f`~wh s
m
~w3fD+,
v m Xs
k
~_`W"u^E~f
T
~w a7wu
!
Y%a7wu~f`_rW"wu~wh s m
~wtf#" +,$%D+&[,¢½A
T W k
W" +,Xa±g
WK
k
W"a7fiW
k
f T a7w
'
Y
~_ruhW"w
m
fiW(ua7_
D+&[,)(*" +,
×Bmtk
~w_`f`a7wBvWî~w
~ 3b
k
WIY3½B~w3f`W
k
}HaoqI~_w
m
fiW(u
Y+(Y, ½
T
W"wvW~f
W 7~w_a7fEs
m
~wtfîYîa7wu^W(wu_Eaofs
m
~wtfîY,<X
m
u)7N q m w f T
Wv~
k
vW3½_i~wvWD+*q ,$ )
SET
W~wtfiW ki
W k W"wBvW
k
a7s
T
+L , N +L-(/.1,
mtk
v m w10~vf
k
a3s Tk
W"_ºbhfi~wh
ûkim
X f T
~_
a7X~g~_
v m w_`f
k
bvfiW"ua7_
m
m
_"®5f
T
WE}7W
k
fi~vW(_v mtk`k
W"_ºs
m
wu%f m f T
W[~wtfiW
k
}Ha7_a3wuf
T
WEW(uh 7W(_a
k
W6f
T
Wv
m
bshW"_
m7
~wtfiW
k
}Hao_f
T
a7f
m
}7W
k
a7sea kim
bwu f T Wv~
k
vW7½h~9oW1~w
~ 3b
k
Wîz[+-a ,la7wu?+-v=,
2
Ww
m
uhW
whWEf
T
WEbw k`m
~wh 1f
k
a7w_
m3k
Xaof`~
m w
W"f²
W_
m
XWs
m
_i~fi~}7W~w3f`W 3W k"32
Wv
m
wB_i~uhW
k
f T W²
uBbsh~v"aofiW(uv~
k
vWîE~f T s m
~wtf`_
ûkim
X4f m ²
'
Y
lm3k
W"a7v
T
~wtfiW
k
}Hao5 N
6(7"
½lWKv
k
W(aof`W
² ~wB_ifºa7wvW"_8+X*, N
&O9(*"O T W k
W-=OMN +:³ +X
Y=,
,X
m
u²
a3wu;"O4N<=Oî³="
SET
W
² bw
k`m
W(u
k
a7s
T BA
~_f
T
W~wtfiW ká
W k W(wvW
k
a7s T0mo
f T W k W(_`bhfi~wh
a7X~g¯®
BA
N
+L
A
,E
T W k W
A
N?>@*+X5,/AB a7wBu Y'C XDC ²
%SET
WWca7XshW%~w
~ 3b
k
WW%_
Thm
_f
T
W~wtfiW
k
}Hao
a7X~g m7
~ 3b
k
WYIbw k`m
W(u f Tk
W"Wfi~X%W(_
prw bw
k`m
~wh uhW
k
W"W%W"dbaof
m
Yev m3kºk
W(_`s
m
wBu_[f m f T W
mtk
~ 7~wa7
~wtfiW
k
}Hao
a3X~g7½ 8E
N
p v m m3k
~wh mo;
~_ra7wa3_`_`~ twXW"wtf îm7
_ m XWKv
m m3kF
+,G f m
W"a3v
T
~wtf`W
k
}±a75_`bv T f T
aof
fF
m%k
Waof`W"uI}3W
k
fi~vW(_
T
a±}3Wruh~ AW
k
W(wtflv m
mtk
_
+&H(JI ,-.GK
+&[,LN
+MI ,
N2
WrWf E +
,luhW"w
m
fiWf T W
v
TBkim
Xa7fi~vrwdbX
W
kmoHE
+ ,6~_f
T
WX~w~XaoÓwtbBX
W kmo
v m mtk
_whW"W"uhW(u
m3k
v m mtk
~wh
>t~wvWlWa7
m
P~wtfiW
k
}Ha7__ºsa3wwh~wh eX m3k
Wf
T
a7w
m
whWîf`b
k
wa kim
bwu fT WKv~
k
vW7½~frXa±g
T
a7sBsBW(w
f T aof
~_rw
m
fv m
mtk
a
W
W"v"a7b_`WK_
m
XW~wtfiW
k
}HaoO^Xa±g
m
}7W
k
a7s ~fº_iW"
½$+&H(/[, .
#
wf
T
a7frv"a7_`W
E +
,K~_bBwuhW
whW"u
QP[m
6W}7W
k
~f~_vW"a k f T
a7f
mtk
² k W(aofiW
k f T
a7wf
T
WXaHch~XKbX wdbX
W k%m7
k`m
bBwu_
m7
~wtfiW
k
}Hao_Ef
T
W
k
a7s T@
W"v
m
XW"_v m
m3k
a
Wî_i~wvWKf
T
WK_`~(W mo
f T WKv~
k
vWî~_[²
a3wuf T W
_`~(W mo
~wtf`W
k
}±a7_a3wu^~wB_ifºa7wvW"_+X5,«½5YRC XSC²
k
W(Xa7~wB_6v
m
w_`f`a3wtf6W"dbaoVf
m "
p lW"
tw
m
w1
m
6W kÓBm
bwu
m
wf
T
W6v TBkim
Xa7fi~v<wdbX
W kÓmo
v~
k
v"bha ká
a k v
k
a7s
T
_~_f
T
W ©,T7¤¦ I©
V¤Lo£¦¬%°
mo
f T W
a7X~guW
whW(u a3_6f
T
WîXaHch~XKbX wdbX
W k6moÓm
}3W
k
a3ssh~wh ~wtfiW
k
}Hao_aofEa7wtges
m
~w3f
m7
fT W^v~
k
vW
M#
fK~_vW"a k f T
a7fKf
T
WE~uhf T °
A:m7
f T W ²ObBw kim
W"u k a3s
TJBA
~__if`~6°t½Þ° N ° AA
2
~f Tm
bhf
m
_º_
mo
7W(whW
k
ao~fFg7½dlWîE~a7_º_`bBX%Wî~w^f T W k
W"Xa7~wh~wh mo
f T WKsa7sW
k f T
aof[W"a7v T s m
~w3f m7
f T W%v~
k
vW~_rv
m
wtf`a7~wW"u~w0Wca3vfig^°I~wtfiW
k
}Ha7_
T#
w m f T W k m3k
u_½f
T
W%E~uf
T
~_v
m
wB_ifºa7wtf[a k`m
bBwu
f T W v~
k
vW
:SET
~_%~_Ka7v
T
~W"}7W"u
g a7uu~wh
vf`~f`~
m
b_
m
whWef`~XW^bwh~f%~wtfiW
k
}Hao_Ka kim
bwuf
T
Wv~
k
vW
T
W"wwhW"vW"_º_`a
k
g3½d~9oW1~w3f`W
k
}Hao_E
mtk
Y7Y1~w
~ 3b
k
WY
# f T
a3_
WW(w a7
k
W"a3uhgs k`m
}7W"w W_iW"
T W k W
½<q=f
T
aoff
T
~_
m
lW k%m
bBwu~_a7v
T
~W}Ha
W^
T
W(w
bBw kim
~wh f T WI
k
a7s
T
®RUt²(
E + BA
,(N °
W#
wJ>tW(vfi~
m
wq^lWv T a k
a3vfiW
k
~(W%f
T
WeuW
k
WW(_²
mtk
T
~v
T
BA
~_vghv~vaogv m
m3k
a
WE~f
T
°Kv m
mtk
_"½da7_llW"Va7_6f
T
WwdbX
W k6mo
v m m3k
_lwhW"W"uhW(u
mtk
vghv~v"aog
v m m3k
~w
RV7H#
wY>tW(vf`~
m
w6Wîs kim
}3Wrf
T
aof
mtk
²a
k
3W1W"w
m
bh T ½ E +
BA
,CO°E³ Y
ö5ö
¤¦¨©o¨i©H¨ ¨`©¢ª¬ ¨«¤¦¥d§^©o¥Ó¥B¨ פ¦¥d§
3
7
1 7
1 8 7 1
1 1 1
1 3 4 6
7
12
10
9 2
11 5
8 3 1 13
3
7
1 7
1 8 7 1
1 1 1
1 3 4 6
7
12
10
9 2
11 5
8 3 1 13 1
2 3 7
6 4
8
9
10 13
12 5
11
1 2 3 4 5 6
0
~ 3b
k
WY7®
#
wtfiW
k
}Hao
a3X~ga7wBu^XW"Wfi~wh I
k
a3s
T
a3wu uW"v
m
Xs
m
_iW(u
k
a3s
T
z ¬d¨«¤98«£*¤¦¥A¡¤98¡¥º¡¤98*K ±©o¤¦¥¡2©H¤¦£¨h¥ ©H¨«
1 1
1 2
1 3
6 2
4 3
13 3 6 3
4 1 13 1
6 1
4 2 13 2
2 1 1
9 10 1 1
11 2 2
9 2 10 2 11 2
2 3 9 3
10 3 3 11 7 1 1
3
8 2 5 2
12 3 7 2
3 2
8 3 5 3 12 1
7 3 3 3 8 1 1
5
12 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0 color 1
color 2 color 3 color 4 color 5 color 6 color 7
~ 3b k W %h®
G w kim
W"u^a7wBu v m m3k
W"ue~wtfiW
k
}Hao
a3X~g
" " % E
prX
m
wh 0s
m
_º_i~
WIv m
m3k
~wh 3_"½56Wv
m
w_`~uhW kk
_`fîf
T
Wva3_`_
m7
KIפLY
¨«¤¦¥d§8îf
T
aof
T
a±}3Wf T W
m
m
E~wh :s k`m
sW
k
fFg¯®IW"f
Wf
T
WwdbX
W km7
v m mtk
_%b_iW(uV½<f
T
W"w f T W k
W^Wch~_ifº_%asW
k
XKbhfºaofi~
m
w
m7
>3Y+(( %_ºbv T f T
aofrf
T
WW"wtf`~
k
Wv m
m3k
~w va3w
WuhW(ubvW(u
ûk`m
X f T Wv
m m3k
~wh mo
f T W mtk
~ 7~wao
a3X~g m7
~w3f`W
k
}Hao_
g¯®
+*+X³´Y,U,N$+
+&+X5,,U,
mtk
YC XSCO²
Ya7wu
+&+iY,U,N$+
+&+û² ,U,U,
SET
~_~Xsh~W"_f
T
a7f!
A
N=#"
ghv~vv m
mtk
~wh 3_v"a7w
Wrv
m
wt}7W(wh~W(wtfig%_`f`buh~W"uIbB_i~wh f
T
WK I¡¢¡£*¤¦¥t§1§7¨`©¢ª¬
m7
f T
W~wh~fi~ao
a7X~g
mo
~wtfiW
k
}Hao_
SET
WrXWW"fi~wh %
k
a3s
T
~_la%uh~
k
W"vfiW(uI
k
a7s
T
uhW
wW"uea3_
m
m
_
p[_6~wIf
T
Ws
k
W}d~
m
b_
_`W"vfi~
m
w6WKa3_`_ºbXWKf
T
aofrf
T
WE~uf
T
~_rv
m
w_`f`a3wtfra k`m
bBwuf
T
Wv~
k
vW
rSET
Ww
m
uhW"_
mo
f T
WX%W"Wf`~wh
k a3s
T a k
WEf
T
W~wtfiW
k
}Hao_"½3a7wBu%f T W k
W~_<a7wIa kv
WfF6WW(wf
T
W[w
m
uW"_
k
W"s
k
W(_iW(w3f`~w ~wtfiW
k
}Hao_Ka3wu-I
T W"whW"}7W
k
"D+,X
m u N I+/I ,X
m u 5E
b k f T W k X m3k
WaI6W~
T
f~_[a3uuhW"uf
m
W"a3v T w m
uhW7½Wdba7f
m
f T WW(wh 7f
T
D+&[,
"D+, m7
W"a3v
T
~wtf`W
k
}±a7
~ tb
k
Wz_
Tm
_laXWWf`~w %
k
a3s
T
a7wBuav~
k
v"bha ká
a k v
k a3s
T
bh~f
ûk`m
X f T
W_ºa7XW1~wtfiW
k
}Ha7
a7X~g
Wfb_
k
_`fv
m
w_`~uhW
k
av~
k
v"bh~f%$
mo
f T
WXWW"fi~wh
k
a7s T5SET
Wîf
m
fºaoV6W~
T f m7
~f`_Ew
m
uhW(_~_Ea
X%bhfi~shW mo
f T W%W(wh 7f
T mo
f T WKv~
k
vW7½_ºa±g°'&
52
WKao_
m
vao°'&Of
T
W%lW"~
T f mo
f T W%v~
k
v"bh~f
#2
6WrbBw kim
Af
T
W[
k
a7s
T ² f`~XW"_"½aX%bhf`~sW mo
°'&½df
T
W(we6Wrv"a7wev m
m3k
f T W[~wtfiW
k
}Hao_6v m3kºk
W"_ºs
m
wuh~wh
f m f T Ww
m
uhW(_
mo
f T Wîv~
k
vbh~fEE~f
T ° & v m mtk
_a7_
m
m
_
2
Wvghv~va7gea7_º_i~ 3w v m mtk
X f m f T WFX)(+*
~w_`f`a3wvW m7
f T Wev~
k
v"bh~f
g0f`aD3~wh ~wtf
m
a7v"v
m
bwtff
T
Wv
m
stg0f
m T
~v
T
W"a3v
T
~wtfiW
k
}Ha7
W"
m
wh 3_îf mB
SET
W~wtfiW
k
}Hao_Ea7_º_`~ twhW"uf m v m
mtk
Xv"a7w
W"
m
wh If
m
av
m
s3g uh~ ¯W
k
W"wtf
ûk`m
X v m
stg X^a7_~w
~ 3b
k
Wîq
m f`Wf
T
aofE° & ~_a7_
m f T
WîE~uhf T^m7
f T
~_~wtfiW
k
}Ha75_`b
h-
a3X%~g
#2
6W%v
m
w_`~uhW k f T
W_ºb
k
a7s T $
mo;
v m Xs
m
_`W"u m7
fT W~wtf`W
k
}±a7_
mo
f T Wv~
k
v"bh~f"½¯lW m3
fºao~w
f T W
m
m
E~wh s k`m
sW
k
fFg¯®
ö5ö
¤¦¨©o¨i©H¨ ¨`©¢ª¬ ¨«¤¦¥d§^©o¥Ó¥B¨ פ¦¥d§ )
¶d»
¶
E + $ A
,N °'&.
¨¡º©Dº¬0 ×£-¤ª¡[²
.° &
»5» SET
Wîv m
m3k
~wh %uhW(_`v k ~
W"ua
Bm
}3Wr~Xsh~W(_lf
T
aof E +$ A
,C° &
>t~wvW1° & ~_Eao_
m f T
W1E~uhf
T
m7
$½° & C E +$ V
,
# w
~ 3b
k
WKqB½6W%v"a7w0_`WW%aXWW"fi~wh
k
a3s T v m
Xs
m
_`W"u m7
av~
k
vbh~f m7
lW"~
T f %
SET
aofr~_rlW
v"a7w0v m
m3k
~fE~f
T
%v m
mtk
_"½
T
W"w0~f~_1ubs~va7fiW"u?%IXf`~XW(_rE~f
T X
îSET
~_
3b
k
W_
Thm
_
a7_
m f T
Wv m
m3k
~wh
m3k
X NxYîa7wuYX N %
b
c d
1 2 3 4
0 a
b
a c
d
2 1
3 2
a 1 b 1
c 1 d 1
a 2 b 2
c 2 d 2
!
"# $% &' ()
*+
,,- ../ 001
223 45
667
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a
b c
d a
b c d
3 3
3 3
4 4
4 4 a 1
b 1 c 1
d 1
a 2 c 2
d 2 b 2
889
::; <= >?
@@A BBC DDE FG
0 1 2 3 4 5 6 7 8
color 1 color 2 color 1 color 2
~ 3b
k
WIq®Kprw ~wtf`W
k
}±a7
a7X~ga7wu~f`_KXW"Wfi~wh
k
a3s TVSET
Wev m
m3k
W"u:~wtf`W
k
}±a7
a7X~gbw k`m
W(u
fF
m
a7wu
m
b k fi~X%W(_
m
î½3Wfb_6a7_º_`bXWf
T
aof6f
T
WrXWW"fi~wh %
k
a3s
T
va3w
W1uhW(v
m
Xs
m
_`W"u~wtf
m
a%_iW"f m7IH
v~
k
v"bh~fº_
m7
k
W"_ºsW"vf`~}3WK6W~
T
f`_°
E
(i°KJ6( (i°KL
NM
g0v m
mtk
~wh f
T
W~wtf`W
k
}±a7_v mtk`k
W"_ºs
m
wuh~wh fm fT WIv~
k
vbh~f`_1a7_
s k W"}t~
m
bB_igIWcshao~whW"u5½hlW mt
f`a7~w f T W
m
m
E~wh
k
W(_`bhf
¶d» ¶ PO
¡£1 º¡©T.©o %¤K
.¥£2¡¨ H©8)Þ£¬¡Fh¥B¨
¡-¤¦¨©H¨i©o¨§o¨`©«ªh¬
A
¤98 KIפL¢©K
° U
¨`©+2¡VÓ¬B¡¨`¡² NRQ
TS
+-°
E
(i°UJ6(°TV ,) VÓ¬¡¨`¡î°
E
WX×°YVO©H¨`¡£¬¡ VÞ¡¤§±¬d£8
.£¬¡ ¤¦¨h¤¦£L8
.£¬¡
3¡-
[ª
8¡- ¡¢¡£-¤¦¥d§%§7¨`©¢ª¬ h¤×£;±ª
¥ Z
»5» k`m
X
SET
W
m3k
W"X Y7½d6W(dw m f T
a7fl~
6WbBw kim
¯W(a7v
T
v~
k
vb~f mo
lW"~
T
fl°K[aXKbhfi~shW m7
° [ ½hf
T
W"w lW1E~5wWW"u ° [ v m m3k
_
mtk
f T W"_`W1~wtf`W
k
}±a7_
>
mI
gebw k`m
~wh av
m
XX
m w XKbhfi~shW m7
a7
¬d¨«¤98«£*¤¦¥A¡¤98¡¥º¡¤98*K ±©o¤¦¥¡2©H¤¦£¨h¥ ©H¨«
f T WlW"~
T
fº_
m7
f T W1v~
k
vbh~f`_
mo
f T W
k
a7s
T
½t6Wra
k
W[_ºb
k
W[f m v m
mtk
E~f
T
°Kv m
mtk
_
ÓSET
WW"a7_`fÞ~wtfiW" 7W
k
T
~v
T
ao
m
_f
T
~_Ev m
m3k
~wh %~_E²N Q
TS
+û°
E ( ,(i°YVD,
SET
W^Wch~_`fiW(wvW mo
a uhW"v
m
Xs
m
_i~fi~
m w ~_~wuBbvW(u
g_ºsW"v~
vs k`m
sW
k
fi~W"_
m7
f T
W0X%W"Wf`~wh k a3s
T
f T aofa
k
WeWchsa7~whW(u ~w
J'
wW~w3f`W
k
W"_`fi~wh :s kim
sBW
k
fFg ~_Kf
T
aofW"a3v T v m
wwhW"vfiW(uv
m
Xs
m
whW"wtf m7
f T
WXWWf`~w
k
a7s T v m
wtfºao~w_aofIW"a7_`f
m
whW
P
a3X~f
m
wh~a3w v~
k
vb~f
SET
~_I~Xsh~W(_f T W
m
m
E~w
v
m3k`m
a
k
g¯®
» » *&*ûh ·
.1£¦¬B¡ I¡¢¡£*¤¦¥t§%§o¨`©«ªh¬
.
¤98F
¥¥A¡-£2¡- <£¬¡¥
E + RV
,HN´°
SET
~_
T
a3ssW"w_1f
m
3~}3WIa^v
m
wt}7W(wh~W"wtfbBssBW km
bwu
mtk
f T WIbw
k`m
~wh uhW"
k
WW
T
W"w
mdm
t~w
mtk
mdm
u uW
k
WW(_
m3k
a 7~}7W(w wdbX
W kmo
v m m3k
_
SET
W%s kim
sBW
k
fFg a
m
bhf
P
a7X~f
m
wh~a7wv~
k
vbh~f`_
T
a7_a7_
m
aIwh~vWKa3ssh~v"aofi~
m w ~wv
m
XsbfiW k a k v T ~
f`W"vf`b
k
W7®Óf T W
k`m
fºaofi~wh
k
W" 7~_if`W kÓ
WB+ F,Ó~_a
k
W 7~_`fiW kÓ
Wf
T
a7f~_}d~
k
fºbaogK_
T ~
f`W"u
m
whW
m
vaof`~
m w
a T W"a3u
T
W(w:whW"W"uhW(uV½5fFghsh~va7gaof1W"a3v
T
~f`W
k
aof`~
m w mo
a
mdm
s p[
m
va7fi~wh a _`Wf mo
}Ha
k
~a
W~
W
f`~XW~w3f`W
k
}Hao_1~wtf m a
k`m
f`a7fi~wh
k
W" 7~_if`W
k
a7X
m
bwtfº_1Wca7vfigf m^
wuh~wh 0a
P
a3X~f
m
wh~a3w:v~
k
vb~f1~w
f T
WXWW"fi~wh
k
a7s TVNP
W"wvWîf T W
m
m
E~w _`f
k
aof`W 7g~_Eb_`W"u5®
~ f T
WXWW"fi~wh r
k
a3s
T
~_v
m
wBwhW"vfiW"u5½o°
k
W 7~_`fiW
k
_a
k
W6_`bIv~W(wtf
mtk
f T Wao
m
va7fi~
m w m
wKa
~
~f<~_Þw
m
fÞv
m
wwhW(vfiW(u°Ó³Y
k
W 3~_`fiW
k
_a k W k
Wdbh~
k
W(uV½
ga7uu~wh
m
whW k`m
bwBu m7¯
vf`~f`~
m
b_Þbw~f
fi~XWî~wtfiW
k
}Ha7_
m3k
Wca7XshW7½tlW[v"a7wIa7
m
vaof`WEf
T
Wr~wtf`W
k
}±a7
a7X~g m7¯
~ 3b
k
WYE~f T )
kim
f`a7fi~wh
k
W" 7~_if`W
k
_~9oW
f T
~_®
~wtf`W
k
}±a7Y7® E ( J
~wtf`W
k
}±a75q® J (
~wtf`W
k
}±a75®
~wtf`W
k
}±a7
" ® ½
~wtf`W
k
}±a7Y7Y3®
~wtf`W
k
}±a7 %h®
~wtf`W
k
}±a75®
~wtf`W
k
}±a7Y,B®
~wtf`W
k
}±a7Y&%®
~wtf`W
k
}±a7 )d®
ö5ö
¤¦¨©o¨i©H¨ ¨`©¢ª¬ ¨«¤¦¥d§^©o¥Ó¥B¨ פ¦¥d§
~wtf`W
k
}±a75z® ½
~wtf`W
k
}±a7 B®
~wtf`W
k
}±a7Y"qB®
% %
# w f T
Wa
_iW(wvW m7
a
kim
f`a7fi~wh
k
W" 7~_if`W k
W7½°î³ YXa±g
Wf mdm
a
k
3Wea7w bw k`m
~wh 0uhW"
k
WW
> m
f T Wa7fifiW
klk
W"_ºbhflX%b_if
W k W
wW"u
SET
W
m3k
W"X Yrs k`m
}7W(_<f
T
aof6f T W k
Wao6a±gh_<Wch~_`f`_l²^_ºbv T f T
aof BA
~_î°
v m m3k
a
W
#w m3k
uhW k f
m
wBu f T
W_ºXaoW(_ifs
m
_º_`~
We²½Ó6WIf k g f m
uhW(v
m
Xs
m
_`WIf
T W X%W"Wf`~wh
k a3s
T
~wtf
m
a7_EXa3w3g v~
k
v"bh~fº_Ea7_s
m
_º_i~
W
W"v
m
Xs
m
_i~wh Ia7X
m
bBw3fº_6f m
c^a
P
a7X~f
m
wh~a7w^v~
k
v"bh~f
~wW(a7v T v m
wwhW(vfiW(uv
m
Xs
m
wW"wtf
p }Hao~u uW"v
m
Xs
m
_i~fi~
m w ~_KuhW"fiW
k
X~whW"u
gv Thmdm
_`~wh 0a0_`Wf m7
w m w~w3f`W
k
_iW(vf`~wh v Thm3k
u_E~wf T W P
a3X~f
m
wh~a3wv~
k
v"bh~f`_
mo
f T Wv
m
wwhW"vfiW(uv
m
Xs
m
whW(wtf`_
>dbv
T
a3w
T
W"b
k
~_`fi~v~_6uhW}3W
m
sW"u
b k f T W k
~w
~ tb
k
WKYrs
k
W(_iW(w3fº_la3wIWca3XsWr
T W k
Wra%X%W"Wf`~wh
k
a7s
T
~_
uW"v
m
Xs
m
_iW(u f T a7w5t_[f
m f T
~_
T
W(b
k
~_`fi~v
#
wf
T
~_rva7_`Wî°N )h½a7wBulWK_ºbv"vWW(uhW"u~wuhW(v
m
Xs
m
_`~w
f T W^
k
a7s
T
~wtf m f
Tk
WW v~
k
vb~fº_
gf`aDt~wh 4:v Thmtk
u_"½
T
~v
T
~_Kf
T
W^Xaocd~XKbBX6W v"a7w u
mB 2 W
mt
f`a7~w $ E N >=% (` ("Y, (Y7Y m7
6W~
T
f +$ E , N Y7½ $ JBN >tY6(`z (U (Y"q m7
lW"~
T
fF+$ J ,;NPqB½a7wBu
$
N>)9(`q (º ( "
(Y=%) mo
lW"~
T
f +$
,HN q > m
lW1XKb_`fbw k`m
Q TS
+`Y6(ºq (`q ,$N´q%fi~X%W(_6f mm3
fºao~w
a3w~wtfiW
k
}Hao
a3X~g v m m3k
a
WrE~f
T
)%v m
mtk
_
SET
Wv m
mtk
~wh ~_E_
Thm
w ~w
~ tb
k
W7%
N2
W T a±}3W7®
v m mtk
Y7®Y
E
(`z J (U 6(Y(q
v m mtk
%h®YUJ6(`z (U
E
(Y(q
E
v m mtk
q®Y 6(`z
E
(U J6(Y(q J
v m mtk
B®%
E
(`
E
(Y,
E
("Y7Y
E ( % J (` J (Y@ J ("Y7Y J ( % (` (Y@ ("Y7Y
v m mtkT"
®<)
E
(`q
E
(` J ( " J (Y=%
v m mtk
z®<) J6(`q J (` 6(
" (Y=%
E
v m mtk
)d®<) 6(`q (`
E ( " E (Y=% J
# w v"a7_`Wew
m
~wtfiW
k
}Hao
m
}7W
k
a7s_1~f`_`W
½f
T
~_
T
W"b
k
~_if`~v
T
a7ssW"w_%ao_
m f m W b_`W
b
m3k
uh~
k
W(vfig
v m m3k
~w Ef T W
m3k
~ 3~wBao
a3X~g mo
~wtfiW
k
}Hao_½Ha3wuî 7~}7W(_Óa3wî~Xs kim
}3W"XW"wtf m7
f T WbssW
km
bwu
m3k
f T W
v
TBkim
Xa7fi~vwdbX
W krmo
v~
k
vba ká
a k v1
k
a7s
T
_ 7~}7W(w
g S
bv7W k
YI
1SET
~_rwWxbssW k1m
bwBu
Y7Y
~_6Wdba7Af
m
%o°
ZH
T W k W H
~_6f
T
WîwdbX
W klm7
v~
k
vbh~f`_6
T
~v
T
uW"v
m
Xs
m
_iWf
T
W
k
a7s TV<'[
v m b k _`W7½
f T WX
m3k
W1f
T
Wî
k
a3s
T
~_EuhW"v
m
Xs
m
_`W"uV½f T W
Wfif`W k f T W
Bm
bwu
% !
ghv~v0v m
mtk
~wh t_Ia
k
W0a
k
W(_if
k
~vf`W"u´va3_`_
m7
v m m3k
~wh 3_
m7
bBw kim
W"u´v~
k
vba ká
a k v^
k
a7s T _
p[_a
Xa7fifiW k[m7<
a7vf½5_`~wBvWf
T
Wbw k`m
~wh ^uhW
k
W"WK²
m3k
v m m3k
~wh
m w v m
mtk
_rXKb_`f_ºaof`~_
g A N )"B½
f T
~_r~Xsh~W(_rf
T
aof1f
T
WuW
k
WW
m
bwBu XKbB_if
Wf T W
mtk
uW krmo
_m XWsW
k
XKbf`aof`~
m w m w WW"XW"wtfº_
Y@ ¬d¨«¤98«£*¤¦¥A¡¤98¡¥º¡¤98*K ±©o¤¦¥¡2©H¤¦£¨h¥ ©H¨«
7 8 9 10 11 12 13 14 15 16
5 10 15 20 25 30
Chromatic number of the circular-arc graph
Number of copies of the intervals 1
2 3
4 5
1 2 3 4 5
0
~ 3b
k
W(B®
#
wtfiW
k
}Hao
a7X~g a7wu^v Tk`m
Xaof`~vwdbX
W k
}Ha
k
~aofi~
m w mo
f T
Wî~wubvW"u^v~
k
v"bha ká
a k vE
k
a3s
T
P
W"wvWIf
T
Wes
k
~XW
a3vf m3k
_
mo
² X%b_`f
WIW"_º_1f
T
a3w
pr_îav
m
bwtfiW
k
sa
k
f"½Vf
T
~_î 7~}7W(_1a7w bssW
k
m
bBwu
m
w%f
T
WX~wh~X%bXuhW
k
W"W6whW(vW(_`_ºa k g
m3k
a3v
T
~W"}d~wh îa3w
v m m3k
~wh
5m3k
~w_`f`a7wBvW7½3~
N ½
6W7dw m f T
aoflWîv"a7w
wu^²W(_`_f
T
a7w:Y
"
½hf
T
a7f6~_f
T
WXaoch~Xao5uW
k
WW mo
asBW
k
X%bhf`a7fi~
m w m w
W"W(XW"wtf`_
#2
Wa
k
WKw
m
fa±6a kW
m7
ae 3W"whW
k
a7
mtk
X%bha
m3k
v m
Xsbfi~wh f
T
WXaocd~XaouhW
k
W"W +û°I,
m7
a sW
k
XKbhfºaofi~
m w m
w°WW"XW"wtfº_
SET
W k We~_"½
Thm
6W}3W
k
½f T W
m
m
E~wh :a7_`gdXshf
m
f`~vIW(_if`~Xa7fi~
m
wV®
+ ,HN "
E
E
Y,
" % E %"
SET
W k Wa
k
WuhW"
k
WW"_
mtk
T
~v
T
a7w%°
v m m3k
~w rWcd~_`f`_
bhf<a1vghv~v6°
v m mtk
~wh 1u
m
W"_w m f
1#
w
~ 3b
k
Wz½
a q v m m3k
~wh Wch~_`f`_
bhfw
m
vghv~veq
v m mtk
~wh
^SET
~_
a7X~g~_bw k`m
W(u
"
fi~XW"_"½
bhf
"
~_%w m f
aX%bhf`~sW mo
q^w m3k
%½5
T
~v T a k
Wf T W
_
mo
f T W6W~
T
f`_
m7
f T W m whgfF
m
uW"v
m
Xs
m
_i~fi~
m
w_
s m _º_`~
W7½wBa7XWg >( îa3wu > +(J>
# w f T
~_E_`W"vf`~
m
we6Wv
m
w_`~uhW
k
7W(whW
k
ao¯v m
mtk
~wh t_la3wu
k
WaHcIf
T
W1s kim
sBW
k
fFg m7
vghv~v~fFga7_º_`bBX%W(u
~w^f
T
Ws
k
W"}t~
m
bB_E_iW(vfi~
m w
!rW"whW
k
a75v m
mtk
~wh 3_
T
a±}3W7½ Thm
6W}3W
k
½ha%}3W
k
ge mdm
u^a7_`gdXshf
m
f`~vîs k`m
sW
k
fFg
#2
² ~_1a
k
3WW"w
m
bh
T
½Óf
T
W(wf
T
Wev~
k
vbha ki
a k vK
k
a3s
T
~wubvW"u
g0f
T
Wbw k`m
W"u
a7X~g ~_a7a±gh_
°[³ Y v m mtk
a
W3½a7_EWchW"Xsh~
W(u~w
~ 3b
k
W7
6SET
W~wtfiW
k
}Hao
a3X~g
T
a7_aIXaHch~Xao5E~uhf
T
W"dbao
f m
½a3wu^
T
W"w^6Wbw k`m
5²fi~XW"_"½BE~f
T
²M\ Y&%½hlWa7a±gh_l 3WfEaIv~
k
vbha ki
a k v[
k
a7s T T
~v
T
~_
<m3k
v m mtk
a
W
SÓm
s
k`m
}7Wf
T
~_s k`m
sW
k
fFg7½(lW
k
_ifv
m
w_`~uhW k f T
Wlv"a7_`W mohm
whWÞ~wtfiW
k
}Haot_`sa3wwh~wh E°v~
k
vbX
W k
W"wBvW"_
m7
fT WKv~
k
vW
2
~f Thm
bhf[
m
_º_
mo
3W"whW
k
a7~fFg7½6WKv"a7wv
m
w_i~uhW k f T
a7ff
T
W%v~
k
vW
T
a7_
m
whg
m
whWKs
m
~w3f
a
WW"u #2
f T W
k
a3s
T
~_1bw k`m
W(u0² fi~XW"_"½5f
T
W(w f T W k Wa
k
W%²s
m
~w3fº_
m
w:f
T
Wbw k`m
W(u v~
k
vW7½
a
WW"uIf m ² Y
p_
m ½
m3k
a 7~}7W"wbw kim
~w euhW
k
W"W²½B6Wuh~}d~uhWK²
g°d®l²@N°[³½B
T W k W
CO°
~ tb k W
"T
W"sB_f
m
~b_`f
k
aofiW1f
T
Ws k`mtm7á
¶d» ¶ !
¡£ º¡î£¦¬B¡$.©H ¤K
rª
8¡-
.
¥A¡(8«¤¦¥d§ ¡î¤¦¥B£F¡¨ H©82ªh©o¥B¥¤¦¥t§1°Y¤¦¨h .¡¨`¡¥¢¡28
.^£¦¬B¡4¤¦¨2¡ ©H¥ ¡£ ¢¡Mh¥B¨
¡-²x£*¤¦ ¡28)K£¦¬B¡¥´£¬¡e² Ch¥¨
¡ ¤¦¨©H¨i©o¨^§o¨`©«ªh¬
BA
¤98
°³ YIU
¨`©6¡%¤.©H¥
¥5KI¤.²@\ °0©H¥ "DC
ö5ö
¤¦¨©o¨i©H¨ ¨`©¢ª¬ ¨«¤¦¥d§^©o¥Ó¥B¨ פ¦¥d§ Y3Y
λ = 4
r+1
µ µ
r = 3 µ = 2 u = 14 = 4x3+2 spaces
intervals
λ intervals
~ 3b k W " ® #
wtfiW
k
}Hao_Esbhf
m
w°[³ Y1a±g3W k _
»5» 2 W
k
_`fÓs kim
}3Wf
T
a7f~
fT
W6v
m
wBuh~f`~
m
w_a
k
W<X%W"f½Hf
T
W"w BA
~_5°h³IY
v m m3k
a
W
5SET
~_Ó~_Óu
m
whW
g
bh~uh~w Kf
T
Wrv m
mtk
~wh Wcsh~v~fig
SET
Wr~w~f`~a7A~wtfiW
k
}Hao~_N (i°
p
fiW
k
bBw kim
~wh
T
a3_
WW(w
sW kim3k
XW"uf T W k
W[a
k
WE²~w_`f`a3wvW(_
mo
~wtfiW
k
}Hao ½ *+I,¢½¯YC C ²Ó½t
T W k W +I,HN
Y+(i°³
Y
×
SET
W
k
_if +û°K³xY=,î~w_`f`a3wvW(_%a
k W vghv~v"aoga7
m
vaof`W"u
m
wf
T
W^°³ Y^v
m m3k
_ SET W"wf
T W
° + , k
W"Xao~wh~wh ~w_`f`a7wBvW"_la
k
Wao
m
v"aofiW(u
m
wf
T
W[°
k
_iflv m
m3k
_
ÓSET
~_Þv m
m3k
~wh XaDoW"_Þ_iW(w_iW
W"v"a7b_`W"Y\a3wu GC Y gef
T
Wv
m
wuh~fi~
m w ²M\°
mtk
W
mtk
Xaogef
T
Wv m
m3k
wdbX
W k
W"uYX¯½5YRC]XSC °%~_Ea3_`_`~ twhW"u f m f T W ~w3f`W
k
}Hao_
+U+
Y=,F°E³ r³ X
Y=,)(FNxY6( J(
+U+
Y=,F°E³³ X5, (N ³´Y+(
<SET
Wa7_`fv m
mtk
°E³ Y1~_a3_`_`~ 3whW(uef
m f T W ~wtfiW
k
}Hao_
+U+
Y=,F°E³ ³ °I,)(N Y6( J(
2
Ws k`m
}7W6f
T
a7ff
T
W[v m
m3k
~w r~_}Hao~uV®
k
_if"½of
T
WE~wtfiW
k
}Ha7_E~f T f T
W_`a3X%Wv m
m3k
u m w m f m }7W
k
a7s
W(a7v Tm
f T W kÞ
W"va3b_`Wf
T
W"~
k
W"wh 3f
T
~_<°a3wuIf
T
Wruh~ AW
k
W(wvW
W"fFlW"W"wfF
m
~w_ifºa7wvWrwdbX
W k _ mo
fF
m
v m
w_`W"v"bhfi~}7Wl~wtf`W
k
}±a7_
m
wKf
T
W_ºa7XWlv m
m3k
~_°
mtk
°³ Y
>tW(v
m
wBuV½Hf
T
W6f
m
f`a7tW"w 7f T v m
}3W
k
W"u
gîf
T
W(_iW
~wtfiW
k
}Hao_5~_Ó²
mtk
f T Wl°
k
_`fÓv m
mtk
_
p[_aXa7fifiW km7h
a7vf½Hf T W
k
_`f5~wtf`W
k
}±a7ta3_`_`~ 3whW(u1f m f T
Wlv m
m3k
X
~_+X5,Va7wu1_`f`a
k
f`_aof3D+&+X5,,NX
Ya7wuf
T
W<a3_if
m
whW~_*++
Y=,F°o³ Ó³BX*,N +û²
°7³BX5,¯a7wBurW(wu_
a7f3" +&+û²
°t³ X5,1N ²Ó³ X Y
5SET
~k uV½
m
w1f
T
W<a3_ifÓv m
m3k
°t³Y<f T W
k
_`f5~wtf`W
k
}±a77~_5+û°t³IY=,¯a3wu
W 7~w_