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(7) b"N.O]LMZcML de f e g.e he igj k l m n oDp q r"sk t q uMk q r i ve wIex y z { | ut } q ~ r t n oDpM hI~ m o j u r i z x yHhI~ m o j u rD.q p q ^ igM~ j uk q F ^ D F ?I k ≥ 2 D C ∼ H £ ¤^ i F k i ¢ K 0 ≤ i ≤ k − 1 ¡ 1,3 £ I . 1 C H ^ D £ I 1 C ^ ? H Fi D£ I 1 C ¥F¦ ¤ t i−1 D£ §H . 3 C ? ¤ T F {t , ti+1 , tk−1 } ¥ © ¨
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(9) ?I ¤.? D i? D © 1D 2 ¤ , . F. .S(1, k) F S(j, k) ¡ j ∈ IF S(2, Dk) ^ "F S(3, £ k) ¥ ª ∪i=k−1 V (C {1, j 2, ¤ 3} ¢ F ) \ T I F S(2, 2p + 1) © p i=0 © I i I« . ¬ D¡ ¤¨ ® ¯ ° ¢ " D I " ≥ 2 M ^ M £ ¤ F S(j, k) ± ¥ F G D ? 2 ² ³ j k n ´ ~Fµ j ¶Dt m n ´ u t j u? I £ ¤ 2· ¥ G ID D H ¤ ¸ F S(j, k) F ¥ ^ )¹ §º" ¹ ¾ ° ¢ 3) ± r n ~ ´ u ¿^¶Fª j n k µ t u ¿ M ».2 · © ¼ ¤ D " ¡ F F® ½ S(1, G I^ M ª . ¥ E(G) ± ? £ HD ^ F ¤^ .^ I . "^ M ¥ DÀ j q ¿ q ~ r.¿ ~ j Á µ ¸ ¥ " F S(j, k) " à ¥ ÄHÅ ÆMÇDÈ É Ê © M^ © IF © © D H ¤ © 2· I D Ë ÌÍ Ì Î ÏÅ Ð) Ñ FÒÓMÔ Å ^ÕDÖ Ñ × ¥ Ñ ÈØ.ÊÙ ^ ¤^Ú ¾ ÛIÜ Ú © ¼ ¤^Ú ¾ ÛÝ ¾ ¥.
(10) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL. . . O WZU.`RSW[ U.O.
(11) "! ! # %$ & ' (*),+- K1,3 % . . . , / ! . 1 ,2 k G 4k 0 ! 3 4 .35 26≥ 2 2 $ 7 89 7 :6 1 k! {xi , yi , zi , ti } 0 ≤ i ≤ k − 1 t i 7 > 9 7 : 8' & 0 C i . ;4 . < = 6C i = 6 . - . . xi , y i zi 0 ! )? 0 #/ . i 3 3! k C = . - ! = . C . 1 Ci ". 3 ! = . )=? " . #@ . i 2 . " i−1 = 1 Ci+1 k≥2 )A .B < C ! 16 = - " . 3 16 26 .3 3 "3 3! i Ci . < ) k Zk Zk. D . ½ D £ ¥ ¥. E#/ . . / < F3 . 16 25 =2 ., ! - -26 ! 0 16 . # * 3 "1 .3# 3 . "1 {xi xi+1 , yi yi+1 , zi zi+1 } i )G
(12) % # B ! H . 3! H . k−1 x1 , . . . , xk−1 } 0 # Y = {y , y , . . . , y } Z = {z , z , . .X. , z= {x} ! 0 , " , 0 1 0 1 k−1 k−1 # - . 3! ! < 1 ! )*I= . *3# * 1 .3 . G T 0 ) {t0 , t1 , . . . , tk−1 } J6" 6 < < . 64 . . = F3 . * 1 0 Ck−1 6 1 ! . - 7 K L 7 M6N 7 9 (7 7 8 ' 8K 1 0 ,O"C)0A . 62 #2" k.≥ 3! 5 .3 # C k−1 " 3 .C 6 . # ! 5 . )<
(13) F S(1, k) F S(2, k) F S(3, k) F S(j, k) j ∈ {1, 2, 3} 26 6 6 1 i=k−1 . 3! # )*? {ti }} . @ .3# 0 = ∪i=0 {Ci 6\ = ! 5j -)*P . k ≥ 3 k) 3 j ∈ {1, 2, 3} * * -FQ3S(j, . 2.< - *R 2 * . Q S T U )-PB*. k F S(2, k) . ""! * . / 5 < F S(3, 2) ! - )*? F S(2, 2) . .< 1*V 3 - 2 . F S(1, . 2) V* * " ! ! <2="! " kW .==2 5 # )=? 5 2 C 3k−1 5 C. 0 @ . . . C0 . % @ < . 1 . . 2B 5 3 3 0 . . 0 C1 x0 x1 . ;2 . . 3 < # . B " # )X6., 3 3 x0 x1 . 0 - 1 " F3x 1 x 0- " .Y7 K L37 y 1 z0 , z 1 x0 , z 1 y0 , z 1 z0 } N 7 9 {x (7 17 x8 0 , x1 y' 08,K x1 z0)*, yP,1 x=02, y 1;y 0 ,# . 7 K L 7 M<N 7 9 (7 7 8 C1 C0 x0 x1 y0 y1 z0 z1 ' 8K ) C0. C1.
(14) OCN TNF[ _ UFR [ S ?ZN.c]
(15) ^U.O W N.[ O[ O W]L% . . E#<! . < . < 36 . .@ 1- 5 . 3 1; 25 . 1- ! . . < . <= F3 . 6 1* 25 32= 5 61 6 . # 0 . . 0 . ! / 25 - ! B C ! . / 1 h ≥ 3 (Ci , Ci+1 , Ci+2 , . . . , Ci+h−1 ) 6 .= 1- . #< " ! - 60 )*? 0 ≤ i < i+h−1 ≤ k−1 2 . 5 C ! . " 1 25 . , Cr ) 0≤p<r ≤k−1 !<1 ! (Cp , ... , . "- !1 " 0! - 0 -! 1 ! *1 p k−1 2 . < # 0 0 . 3 ! " . r * x0 ∈ {x x,p−1 ! ;Cp−1 1 F 1 1 . 0 xp Cp y , z } p = 0 . = # 0 =. 3 ! 6 . p−1 1* k−1 k−1 F k−1 1 ; xr+1 ! )6P,"C r+1 1 ;< Q "0 ! " 1x r. C 3 r! 5. . x0r+1 ∈ {x0 , y0 , z0 } r = k − 1 1 6. 3 ! 5 1 . ) yp zp yr z PB/ 6 @ . 1 26r . , B < / @ . #% 2 ) #3 5 1; 1 6<0 . 3 6 . F S(j, k). -7 9 ' 8K& 7 9 N 7' 37 : 7 $ 9 G ∈ {F S(j, k), j ∈ {1, 2, 3}, k ≥ 2} M ' 9 $ ! 3 8 # L " . 7 / 8 9. 7 ' $ 9 " : 8 K. ' $ 7 M ( '. ' 9 " " & 7 8 L 9 ' 8K' 8 G $#% 2& G\M ' 8K '3& **& M(" 87 ' 8K#" 8 & 2+," 87 " M " & ' 9 7 K
(16) ) 7 : 9 7 ># 8/7 ' $ /$ & ' ( M ( ) 9 7"9 3 : 7 7- " & & " (. 83L/: " 37 : 9 7 M C0 i (i ∈ Zk ). !21 " :7 ) 7 : +. 8 $ " 8 9 ' 8 M7 >3' $ 9 & +" 877 K L3743 " 8 83L9 7$ & ' ( 9" Ci 9 7$ & ' ( i Zk M C 5 !21 " :7 ) 7 : +Bi+1 7)78 8 $ " 8 9 ' 8 M7 >3' $ 9 & +/9 ("/7 K L37 M@N 7 9 (7 7 8 ' 8K i Zk M Ci ' 8K<68 " 87"N 7 9 (7 7 8 ' 8K Ci+1 Ci−1 C !21 " :@7 ) 7 : + " K K i 8 Z M $ " 89 ' 83M/i 5 7 >3' $ 9 & +B9 (",7 K L 7 M/N 7 9 (7 7 8 C ' 8K i ' 8K<68 " 87"N 7 9 (7 7 8k ' 8K Ci+1. Ci−1. 7" : 7 " ) 7 : ( 7 8 5 +- . k. M(" K K. M. Ci $. M ' 9 M *7 M " 8 & +
(17) 9 7 . -6- 1 ; . " 1. ! ) 1 * . ). P roof. = F S(j, k) 8 . . M . F #, . 3 15 G , 2=* " " ! =j ∈ {1, 2, 3} M . 3! 5 @ 1; . . . F ./ @0 2 ) G\M 0 2 C ? 5 / 2 1 F "! 6-5 ! # .@i 3 " 1 Ci G 0 ti !t = . {x , y , z } )#95 . < < F #/ 2M 2 . 3 . "1 3 5 1 . .= . i . . i i= i 6 . ) M 0 Ci Ci−1 ∪ Ci+1 A 1* " = 2 3 5 1 2 . . .@ = .< 3 3 1 2 . . )5A M 1* = C2i 3 3 C 5i+1 1 - 2 . . M ./ C 5i−1 . 3 <C i1 - 2 . . )(M 95 . 2 C i−1 ! Ci ! . 25"! 6 0 " .@ 0 M ..3! " Ci 1* 26C 6i+1 . ) k G.
(18) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL . X5 ! <". 2 = " 5 . #/ . 3 3" 1 2 . . )5
(19) . = F3 6 F # .= 3 35 2 . . M . CF3i−1 . 3 . <C i Q Ci ./ C 5i+1 3 ) < / 2 1 32= 3 ! 26 . 6 0 G k # 66P,< #B @ 1 "< . 1 " 1"9 + 37 M F S(j, k)
(20) . ! 16+- < . / 159 + 7 . ! . ! )<A 1. # 1 < 3 . ! ! 1 2! 2= ; #/9 + 37 . ! . 9 + 37 . ! );P,. * . ! "- 1 1 2.0 * . 1 #3 2.1 . 1- #3- " C ! ) 2.0. 2.1. /
(21) < . % 16 l(P ). - 0 # . l(Γ). !). P. - 0 #B@ # . FU.RO W[ O DLMZLSWaCNWS ][ O U. Γ. ! " . B #. F S(j, k). PB #/ F 16@ ! <
(22) 8 6' 9 7 K . @ . 0 v G 2 .26 . ! 5. 2Y ! 0 3#< . . < 3 . " 6. 2 v . 3! . ,@ . . /4 . G . , F% 16 . - ! 33 0 0 1 // . F, 16 . 2G . ),PB #% @ 0 N (v) G . Bv1 #,B0 9 : ' 8 L '3& ' :7 > 9 7 83M " 8 )@
(23) . - ., " G 0 $ " 8 9 : ' $ 9 " 8< 5: 7 K ' $ 9 " 8< 1 6 . )*
(24) 6. ! "- 6 1- 1 . 6 1 6 . # ) G. µ(G). 6-7 9 N 75'=N 3' : 9 9 75$ ' N $L : ' <' 8K5& 7 9 N 769 76N ' : 9 9 " 8 G {V1 , V2 } " 9 M4) 7 : 9 7 >M 7 9 M M ' 759 ' 9*7 ' $ ,) 7 : 9 7 >
(25) 8<M " 76M ' N M 7 9 M 8 ' 9 7 K 8 9 @ " '9 : ' 8 L & 7$' 8 K& 7 9 0 N 7<9 7"L : ' " N 9 ' 87 K# 8B9 ' 96(W' 1+ ⊆ V 17 8 G $% µ(G) = 0 $ µ(G ) = 5 . @ ./ . 3# . ! 5
(26) " P roof. {V1 , V2 } 5 6 ! " 3 3= ! 5 -) 8 3 0 6 ! G " 3 3< ! < . 3 . 0 - 1 "< G . 3 ! )+- 3 G @/- 1 < . % 1 0 ) # G F 0 1 ! %3#M .. 3 <26 . B . G F/ = . 0 )@+- V1 \ W1< 1 2 ! / . )=X5 ! 0 = " V= . A< ⊆1 V20 = ! / = 0 " V2 6 = ! # = 3 3 . .= .G . 5 . = . 0 . .5 . )
(27) .25. " =0 . |W1 | + 2 0 < .3 3 .-V)2 95 . "! 6 "0 F #@ ." 3 = ./ @ . V 2"\ .A .3 .% 16 BM . < .,0 " " Q . 1 ., G. M.
(28) OCN TNF[ _ UFR [ S ?ZN.c]
(29) ^U.O W N.[ O[ O W]L% . 1 "< . B 1 ) . # , 1 "< . B 1 = @ G ! . C != 1 6< G . 1 0 0 . 2" .@ 5 ! ) G +- ;! ; . 3# *.3! " 1 - 1 ; . * 1 S(j, k) µ1 (j, k) . ! "- 5 1; µ(j, 1 k) . 5 1; #3 . . F! " - 5 1; 1 1 µ2 (j, k) < . 3 5 1; #3- ) 2. 7/ ' ) 7. • µ(1, 3) = µ1 (1, 3) = 9, • µ(2, 3) = µ1 (2, 3) = 8, • µ(3, 3) = µ1 (3, 3) = 6, • µ(1, 2) = 9 5 µ1 (1, 2) = 3,. • µ(2, 2) = 10 5 µ1 (2, 2) = 4,. • µ(3, 2) = 12 5 µ1 (3, 2) = 6 $.
(30) 5 # 6 . . . 6 F3 . 1 6 26 1 . P roof. )0 . 5"- 1 6< . F S(1, 3) x0 , x1 , x2 , y0 , y1 , y2 , z0 , z1 , z2 , x0 M .3 . . < = 3 )*
(31) " 5 2 ! . ! x1 x2 ∈ M )*A . ! 2"!t0 x*0 )*A . ! x 1t1 ∈ 0 M y 0 y1 , t 1 z1 , t 2 z2 , z 0 y2 ∈ M 2 ! " ! ) . ! ) )6A . ! ) <2="! x2 y0 ∈ M . . ! ) x@2 t22=∈"M ! 5 0 )
(32) y1 y2 , t 2 z2 , z 0 z1 ∈ M ! 6 / @ F # 3 . 1 < 0 . y 0 y. 1 , y 2. z 0. , z1 z2 ∈) M 3 E#@ #3 # " 3 . 5- 1 5< . 3 = .3 . . t.0 x0 t 0 y0 3 3 . 6 . 5 . 3. . 1 ) t0 z0 µ(1, 3) = 9 A ; -2 Q3. 2. * ; .< ; F # 1 ;< . 3 ) 6 8 . . 1 ; .Y Y #% . R . F 1 . @ - 1 @< . 0 3 @ 1 .3 ,F S(2, . 3) / 6 )PB< < 26. 2 - 1 "< . 3 "2 .% . . @ 0 F S(2, 3). . . . . 3 . . B " . 2< < 2@2 # = . ! < 3 0 ) .3 "- 1 6< . ! )-95 . µ(2, 3) = 8 . ,1 #B . R . / " ., < 0 . 5 #3 . ! F S(3, 15 3) .-) 8 . K 3,3 F% 1 < K3,3 +- 25 3 6 << 3 # 5 ! 61 ) !) A 6 ! .5 6 < . !µ(3, 1 3) 0 F S(j, 2) j ∈ {1, 2, 3}.
(33) 6 % 768 ' N 7 : M µ1(j, k) " 37 : 7 $ 9 ' 9 $ ! 83L 4M " 9 + 7 1 " F S(j, k) ' : 7=L ) 7 8BN + 0 (j ∈ {1, 2, 3}). • µ1 (1, k) = 2k − (−1)k ,.
(34) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL • µ1 (2, k) = 2k ,. • µ1 (3, k) = 2k + 2(−1)k $. E#+- <<32 . 1 ! W ! C ! . ) k=2 3 µ1 (j, k) #% . 3!k = .Y . . ,2@ ! < PB2 . B ! µ1 (j, k) k !) # 1 2 − 2)
(35) =1 26 . < F S(j, Q26k 5 1 ! k )≥+- 46!j ∈ {1, . 2, 3} 6 51 ! 5 . ! 0 ! )PB6 . 25 . 1 C1 F S(j, k) j ∈ {1, 2, 3} 0 . Ck−2 . /C4 k−1 .% C F3 . < 1 @ F3 C k−1 . C0 0 Ck−2 1 #,@< . / 3 ! ., ! %2 #B < ! . 0 , " C 1 ""! "< = -26 0 0 − 2) . 0 < #5 3 F S(j . ! ,)kA . j ! .3∈{1, 62,. 3} ! "- 6 1- 1 . j 1- #3j 1 26. < " . . .6.3! " 1 G ! . ! . 2 3 . =.3! "1 =F 1;S(j, - k) 1 5< . 3 5 1; #3- 1 F S(j, k) . < . 3 36- 2 . . .1 < = . . 3 36 2 . Ck−2 Ck−1 . ) C1 +- 6C!0 6 W . 5.3! " = 1* 1 6< . 3 5 1* #3 . " ν(e, e0 ) 1 . . < 5 2< 3 3 . 0 )
(36) .@2= P roof.. e. e. a1 = ν(xk−2 xk−1 , x0 x1 ), a2 = ν(xk−2 xk−1 , y0 y1 ), a3 = ν(xk−2 xk−1 , z0 z1 ), a4 = ν(yk−2 yk−1 , x0 x1 ), a5 = ν(yk−2 yk−1 , y0 y1 ), a6 = ν(yk−2 yk−1 , z0 z1 ),. . a7 = ν(zk−2 zk−1 , x0 x1 ), a8 = ν(zk−2 zk−1 , y0 y1 ), a9 = ν(zk−2 zk−1 , z0 z1 ).. P9 ! #2= ) 0 +- ! 60 µ 1 (j, k) = 1 i=1 ai . . . 6 # ! . 0 Ck−1 C0 H . 3! @ ! -). ;P .. j=1. 2= 0 a2 + a6 + a7 = 2µ1 (1, k − 2), a1 + a5 + a9 = µ1 (3, k − 2), a3 + a4 + a8 = µ1 (1, k − 2).. P ! * 1 3 . #26 . . . . . xk−1 y0 yk−1 z0 P roof. " 3 6 1 ) zk−1 x0 F S(1, k) A .B 5 ! 2 @ 3 . a6 + a7 xk−2 y1 yk−2 z1 )"A .B0 52a 2 + 52 zk−2 x1 H H1 = (V (H), E(H) ∪ {xk−2 y1 , yk−2 z1 , *2 . )#* % 1 < . , 16 #3- 1 zk−2 x1 }) F S(1, k − 2) 1.
(37) OCN TNF[ _ UFR [ S ?ZN.c]
(38) ^U.O W N.[ O[ O W]L% . .3 . . 2% 1 < . 3 @ 1" #3- 1 0 . Y 1" #3- F S(1, k) k−2 y1 2 " Hx 1 < 1 . . . 1 . 1 H1 yk−2 z1 )6A 1;1 26 5 @- 1 5< . @ 1* #3- . . . " 3 3 zk−2 x1 ) 1 a2 + a6 + a7 = 2µ1 (1, k − 2) +- <! <. 2 )H
(39) H2 = (V (H), E(H) ∪ {xk−2 x1 , yk−2 y1 , zk−2 z1 }) = < ) #* ,- 1 " . B 16 #3 H2 1 1 . . . F S(3, .k<− 2) 1 "< . , 16 #3 1 H2 xk−2 x1 0 1 F S(1, k) 2 " H 1 . 1= #3- 1 . . . . H 1 . 1" #3 1 1. . H . 2 ) y.k−2 C y!1 .3 # 1 H2 zk−2 z1 ) a1 + a5 + a9 = µ1 (3, k − 2) ? ! . 26 a4 + a8 ∪ {xk−2 z1 , yk−2 x1 , = a 3 + < 6 H3 = (V (H), E(H) ) # 1 < . zk−2 y1 }) F S(1, k − 2) 1 #3 1 . . . @ ./ 1 < . Y 1 #3 1 1 H <32 " Yx-k−2 1 z 1<0 . 1" #3 1 .3 . . 1 F S(1, k) . @ @- 1 5< . @ 1* #3- 1 .3 1 . . H3 )6
(40) 3! yk−2 x1 1 H3 zk−2 y1 6 < ) H1. a3 + a4 + a8 = µ1 (1, k − 2). 6;P .. j=2. 2. 2= 0 a1 + a6 + a8 = 2µ1 (2, k − 2), a2 + a4 + a9 = µ1 (2, k − 2), a3 + a5 + a7 = µ1 (2, k − 2).. PY ! * 1 3 . #26 .< . . . . P roof. xk−1 x0 yk−1 z0 = 3 5 1 ) zk−1 y0 2 " 3 . A ./ 5 < !F S(2, k) 0 a6 + a8 xk−2 x1 yk−2 z1 )"A .B 52 a 1 + =2 zk−2 y1 E(H) ∪ {xk−2 x1 , yk−2 z1 , 32H5 . ) H #*1 = -(V (H), 1 . 1; #3 1 1 H1 zk−2 y1 }) - # 1 F -S(2, < k − . 2) 3 ; 13 #3 1 ! 2 1 F S(2, k) a1 +a6 +a8 = ) 2µ1 (2, k − 2) PB . ,. 2 % H2 = (V (H), E(H) ∪ {xk−2 y1 , yk−2 x1 , < 5 ) #* 1 < . 1- #3- 1 zk−2 z1 }) F S(2, k − 2) 1 H2 . 1 *< . 1 #3- 1 . C ! .3 # 0 1 F S(2, k) a 2 + a4 + a9 = ) µ1 (2, k − 2) +- = < H3 = (V (H), E(H) ∪ {xk−2 z1 , yk−2 y1 , zk−2 x1 }) )# , 1 = . / 1 #3 1 = # F S(2, k − 2) . 1 . , 15 #3 1 . 1 C ! H . 3 # 1 F S(2, k) a 3 + a5 + a7 = ) µ1 (2, k − 2) 2.
(41) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL . ;P .. j=3. 2= 0 a1 + a5 + a9 = 2µ1 (3, k − 2), a2 + a6 + a7 = µ1 (1, k − 2), a3 + a4 + a8 = µ1 (1, k − 2).. 8 ! - = = 3 3 6 1 ) zk−1 z0 F S(3, k) A ./ 5 ! xk−1x0 yk−1y02" " 3 . a5 + a9 xk−2 x1 yk−2 y1 )"A .B0 52 a 1 + 52 < zk−2 z1 H1 = (V (H), E(H) ∪ {xk−2 x1 , yk−2 y1 , 2H . ) #* %- 1 . , 16 #3- 1 zk−2 z1 }) 2) .%- F3 . % . F S(3, k 1 − . 15 #3 1 1 3! H1 2 1 F S(3, k) ) a1 + a5 + a9 = 2µ1 (3, k − 2) +- - 2 , 3 )=H X 2 1 5< . @ 1*F S(1, #3 k − 1 2) ./ #@x"k−2 F3 y 1. y k−2 @ .3z 1 zk−2 x1 1 H2 .5 1 . < 1- #3 1 . ) 1 F S(3, k) a2 + a6 + a7 = µ1 (1, k − 2) +- = < H3 = (V (H), E(H) ∪ {xk−2 z1 , yk−2 x1 , zk−2 y1 }) ) #* ,- 1 < . B 1 #3- 1 " .< 1 F S(1, k − 2) 1 H3 0 < . 1; #3 1 . C ! . # ) P roof.. 1. 6 - . µ1 (j, k) =. F S(3, k) P9. i=1. ai. a3 + a4 + a8 = µ1 (1, k − 2) 2. )
(42) .@ 61 26 1 . / < . . µ1 (1, k) = 3µ1 (1, k − 2) + µ1 (3, k − 2), µ1 (2, k) = 4µ1 (2, k − 2), µ1 (3, k) = 2µ1 (3, k − 2) + 2µ1 (1, k − 2).. E# . 3! .µ1 (1, k − 2) = 2k−2 − (−1)k , µ1 (2, k − 2) = 2k−2 , µ1 (3, k − 2) = 2k−2 + 2(−1)k ..
(43) 3! µ1 (1, k) = 3(2k−2 − (−1)k ) + 2k−2 + 2(−1)k = 2k − (−1)k , µ1 (2, k) = 4(2k−2 ) = 2k , µ1 (3, k) = 2(2k−2 + 2(−1)k ) + 2(2k−2 − (−1)k ) = 2k + 2(−1)k ..
(44) 6 . 5 3 1 ). .
(45) OCN TNF[ _ UFR [ S ?ZN.c]
(46) ^U.O W N.[ O[ O W]L% .
(47) {1, 2, 3}). . 7/86' N 7 : )7: * 7M0 %. " ,37 : 7 $ 9. . ' 9 $ ! 8 L M" . µ2 (j, k) F S(j, k) (j ∈ ( 7 8 M" K K"' 8K k " 9 7 : (. M 7 $ µ2 (j, k) = 0 k µ2 (j, k) = 2×3 2. PY . 63 32= 3#+- # ) P . k .- ="0 - µ 1 2 (j, k) = 0 . 1- #3- 1 )? k M 2 F S(j, k) 0 . 3#@+- << 2" < 5 2 #@ 2 . ! < 25 0 6. . )PB 0 3 5 1 4 . . <0 " F3 .C i C i+1 6 1 = 1 .C i 3 C i+1 5 1 2 2 M# * = 3 3 3 * 0 2 . 0 3 2 Ci Ci+1 3 ; .=-* " .! . C !*2 #= ."6 1 -< . = 13 * ! ) 95 . 626 # 6.3! " 6 1- 1 . 6 1- #3 . Ci ∪ Ci+1 ! . k ) 2 P roof.. 2. F S(j, k) j ∈ {1, 2, 3}. ?3 . µ2 (j, k) = 2 × 3.
(48) < .
(49) . 6 % 7/86' ' : 75L ) 7 8BN + 0. N7:M. "2" 3! µ(i, k). . " , 7 : 7 $ 9 ' 9 $ ! 83L M " . F S(i, k) (i ∈. {1, 2, 3}). , 7 8 , 7 8. k. k. M(" K K. •. µ(2, k) = 2k ,. • •. µ(1, k) = 2k + 1, µ(3, k) = 2k − 2,. M"7 ) 7 8. •. µ(2, k) = 2 × 3 2 + 2k ,. •. µ(1, k) = 2 × 3 2 + 2k − 1,. •. µ(3, k) = 2 × 3 2 + 2k + 2 $. k. k. UDL WZRSWRZ N._ ^L. k. R_ W
(50)
(51) U.R W DLMZLSWaCNWS ][ O U F S(j, k). 64 , --7 9 N 7<' 37 : 7 $ 9 ' 9 $ ! 83L2" "9 + 37 " 7 8 M 9 7 ' $ 9 " : ' M<7 > ' $ 9 & +" 87," :<9 ("@$ + $ & 7 M 1 ' 8K@G 7 ' $ = ,F$ + S(j, $ & 7#k) " $% \M G\M ' M2' & 9& 7 ' M 9-" G87( ) 7 : 9 7 ># 8/7 ' $ /$ & ' ( Ci (i ∈ Zk ) $ +- . -"< 1 6 . @ 1* #3-. .. )5+- 5! 5 . = . P roof. M 1 2 1 6 . )*X5 ! 62 ! 1 .G # 5 3 35 1 C. i . . i Zk )
(52) " # = 1 . 1 M C i ti xi G\M 0 xi 3#! . F . 33 6 )6E#@+- 2 =Ci−1 Ci 0 xi Ci+1.
(53) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL. T. 5 .3 . @ ./< # " 1 )6
(54) = 2 3 3 6 . . = . M ! "2 6 " G \ = 4 . . <. y i.3 . z i 4 y it. i z . i . Ci .C i−1 #3- ! )
(55) ! = # C i 5 1 Ci+1 . . . M M y i ti zi 51 1. @ 3 5 )=
(56) 3!G 2\M " 5< 5 2 Ci−1 0 ! - C i . <25 .. C i+1 6 6 @ 2Y" # . 0 # "G# \ M ). # "+- <! < ! ! B " " 1 . <@- 1 < . , 16 #3- .. M = F #/ 2@ # . G = F S(j, k) ! )XH 2 . #2 G \ M 6 1 # Γ 1 0 Γ1 Γ2 Γ2 ' 3 " : C i # ' 3 " : 0! ) 0 Γ1 & Γ2 & . 4-7 9 N 7@'
(57) 37 : 7 $ 9 ' 9 $ ! 8 L" <9 + 7 " M ' $ 9 ' 959 7 ' $ 9 " : M ' M7 >3' $ 9 & +@9 ("/$ + $ & 7 M 17 8 9 G 7=& 7 83FL S(j, 9 M#"k) 9 7 M 7 $ % 5 9 ("$ + $ & 7 2M(& ' ) 79 G7"\M ' M 743' : 9 +' M k 5 ' 8K9 " M 7& 7 83L 9 M<' : 7<K M 9 8$ 9( 7 8 k M(" K K $ +- . . 2, # < 1 )HE#%+- <<,T 1 Γ 15 2Γ # 2 "! 5 = 2 G \ )M+- 5 5. ! "- k1 25 . , < <.3! " 1 Ci" < 4 < 25 )/P,< k2 Γ2 . )PY . 3 ;20 2 + k1 "! 5 l(Γ 1 ) =3 3k @1 .+ k2 .-l(Γ 2 ) = 3k .B . 3 )6
(58) k . . 0 k1 k2 0 k1 0 k2 < 3 . "3 . )=P . = .-2""! = Γ1 . Γ2 0 k1 .- . 3 )
(59) . k. 0 " . . 0 ) 0 k2 0 k2 k1 Γ1 Γ2 0 64-7 9 N 7@
(60) ' 37 : 7 $ 9 ' 9 $ ! 8 L " <9 + 7 " M ' $ M 1 G = F S(j, k) 9 ' 9*9 7 ' $ 9 " : ' M=7 >3' $ 9 & +9 ( "<$ + $ & 7 M ' 8K ' !" M 759 ' 9*9 7 : 7 Γ1 ' : 759 ( "$ 2" & 8 M 7 $ '39 G ) 7 \ M ' 3 " :5$ & ' (;M ' 8K ( 9 Γ2 $ . 7 8 Γ1 & Cj Cj+1 j ∈ Zk \ {k − 1} $
(61) % 9 7 :
(62) 7 M ' 7 : 7 $ 9 ' 9 $ ! 83L 0 " 9 + 37 1 M ' $ B9 ' 969 7 2 & ' $ 9 " : G \ M 0 ' M 7 >3' $ 9 & +9 ( "$ + $ & 7 M 0 ' 8K 0M ' ) 8 L9 7 " &&" . ( 8 / L : " 37 : 9 7 M 0 P roof. . 1 i" < 4 Z k < Γ1 k1 + k 2 = k. . Γ1. Γ2. ! " : i ∈ Z \ {j, j + 1} C M Γ0 & ' 3 " :/ "' 8K," 8 & +
(63) ! C ' 8K C k ' : 7 Γ0 & ' 3 " i : 2 j+1 5 ! l(Γj 0 ) = l(Γ '2 8K 0 )−4 l(Γ ) = l(Γ ) + 4 $ 1. 1. 2. Ci. M. Γ2 &. '3":5. 2. . 5 26 . ) 8 . 6 " 4 6 2H ! " 52 ! 6 6 1; 3 . C #j Cj+1- . C j .Γ 1 . . 0 2 0 tj z j . .M 3 ! 1 Γ1 . xj−1 xj tj yj yj+1 . ! ) 8 .x j−1 . 3 5 " < 4 " . 3x j = C ! j−1 P roof.. xj xj+1. M. Cj+1. Γ1. Γ2.
(64) OCN TNF[ _ UFR [ S ?ZN.c]
(65) ^U.O W N.[ O[ O W]L% . . - < # . . "! < .3 ., . 2 . 0 Cj . yj+1 .C j+1 6 =. - Γ! 1 = 1 . t. j+1 x.j+1 xj+2 . 0 xj+2 M tj+1 zj+1 ! )<I= . # P x j+1 0 Cj+2 ) 0 0 yj+1 yj+2 x x t y y t x x 1 j j j+1 j+1 j+1 j+2 j−1 j 26 .3 . 5 " . . W . 0 0 0 0 Γ2 P2 = zj−1 zj zj+1 zj+1 zj−1 zj+1. < # ) 8 5 5 1 5 1*?- ! = ) +- "! " 1 <1 2 . @ 3 . 1 . < . 1 . % . ),+- xj xj+1 t j zj tj+1 zj+1 M zj zj+1 tj xj tj+1 xj+1 M0 1 6 @ # 0 ! . 1 6 . )5
(66) .@ " ! P1 Γ1 P1 = . " . 6 . !. . 1 . # 0 0 0 0 xj−1 xj xj+1 xj+2 Γ2 zj tj yj yj+1 ?; ! < ! )"PBP 2 ., . 2 " 1 P 2 .3= z . j−1 . 2. 2 0 2 tj+1 zj+1 zj+2 . 0 " 4 25 " . /1 . # 0 . 0 ) Γ1 Γ2 0 " < 4 Cj C j+1 # 0Γ " 2 4 ! 15 . % . #B 1 i Zk \ {j, j + 1} Ci Γ2 Γ1 0 . 2 Γ0 ! 6C i1 4 " 4 # " 4 ! )
(67) 5 . / 1 0 Γ2 Γ1 Γ1 . 2 0 ! . 6 1 ! . ) 1 ! . 26 5 5 . @ 1 Γ2. xj. Γ2. 4. x j+1. yj+1. yj. xj. x j+1. yj+1. yj. tj. t j+1. tj. t. zj. z j+1. zj. z j+1. Cj+1. Cj. D . . Cj+1. Cj. ¥. j+1. M ^ ^ ¤ ½ ¥.
(68) ., B .,+- << < <2 < ,/& " $ ' &9 : ' 8 M " : 0 & ' 9 " 8 " = 9 +. / 7 ) . 64-7 9 N 7@'
(69) 37 : 7 $ 9 ' 9 $ ! 8 L" <9 + 7 " M ' $ G = F S(j, k) ' 81K ' !" M 759 ' 9*9 7 : 7 9 ' 9*9 7 ' $ 9 " : M ' M=7 >3' $ 9 & +9 ("<$ + $ & 7 M 2& Γ1 ' : 79 3 : 7 7< $ " 83M 7 $ '3G 9 ) \7<M$ & ' (;M ' 8K (. 9 Γ28 $ Cj 5 Cj+1 Cj+2 j − 1, k − 2} M ' $ 9 ' 9 ' 8K ' :7 ' 3 " :"' 8K M ' 3 " Z: k \ {k 7 8@ 9 7 : 7( M"' Γ1 & Cj+1 Γ2 & $
(70) % ' M7 >3' $ 9 & +<9 (" 37 : 7 $ 9 ' C9 $ j ! 83L Cj+2 " = 9 + 3 7 M. ' $ @ 9. ' 9 9. 7 ' $ 9 " : 0 G \ M0 ' ) 8 L9 7 1" & & " (. 8 L/: " 37 : 29 & 7 M 0 $ + $ & 7 M 0 ' 8K 0 M Γ1 Γ2 ! " : i ∈ Z \ {j, j + 1, j + 2} C M Γ0 & ' 3 " : 5' 8K " 8& +( C M Γ & ' 3 " : i 2 5 ! C ' 8K Ck ' : 7 Γ0 & ' 3 " :"' i8K C 2 M Γ0 & ' 3 " : j j+2 j+1 5. !. l(Γ01 ) = l(Γ1 ) − 2. ' 28 K. l(Γ02 ) = l(Γ2 ) + 2 $. 1.
(71) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL. . 8 . " < 4 * .< 3 1- 1-+- <<, 5 ! " " 3 t z . x x . @Γ 1 .3 . " C j 0 Γ1 xj−1 xj tj yj yj+1 ! ) 8 . C Γ " < 4 6 = # Γ . . 6j j " 3 jy j+1y )
(72) . M 1 j+1 j+2 2" 5 j+1 .3 2 . 6 5 . 0 Γ1 Q1 = x0j−1 xj tj yj yj+1 yj+2 tj+2 zj+2 zj+3 . . @ ) @ 0 Γ2 Q2 = zj−1 zj zj+1 tj+1 xj+1 xj+2 x0j+3 . - . ) yj+1 tj+1 zj+1 zj+2 +- =! 5- 1 "t1 j+2 x2j+2 . 3 * .M 1 < . " tj zj xj xj+1 . 1 . . zj+1 zj+2 xj+2 tj+2 M xj tj zj zj+1 xj+1 xj+2 )+- . . ! . . , 1 < < . . . Y )
(73) . . . @ ! zj+2 tj+2 0 1 M M Y3# . Y B ! Q 1 0 Γ1 Q01 = x0j−1 xj xj+1 tj+1 zj+1 zj+2 zj+3 ,?- " 1 H # 0 0 0 Q2 ! ! ) Γ2PBB . ,. 2 Q2 " 1 = z j−1 / z j. t j y j. y j+1 . y j+2 2%tj+2 . 2 xj+2 # x j+3 . < 0 . 0 ) < . 2 <. 2 0 " < 4 < 25 . Γ0 1" Γ2 Cj Cj+2 Γ2 Cj+1 Γ1 < 4 )
(74) @ . Y 1 1 ! . <26 . H 1 . " Γ1 Γ2 < 1 2 0" < 1 ! . ),A < 2 < 4 # 0 " < 4 ! 1 . Y i. ∈ #Z1 k \ {j, j +" 1, j4 + 2} C i Γ 2# 0 Γ1 Ci Γ2 0 " 4 ! ) P roof.. Γ1. xj. x j+1. yj+1. yj. j+2. t j+1. zj. z j+1. t j+2. z j+2. Cj+2. Cj+1. D . y. yj. . ¥. x. x j+1. xj. j+2. y. tj. Cj. x. j+2. y. j+1. j+2. tj. t j+1. t j+2. zj. z j+1. z j+2. C j+1. Cj. M ^ ^ ¤ . C j+2. ¥.
(75) ., B .,+- << <2 < ,/& " $ ' &9 : ' 8 M " : 0 & ' 9 " 8 " = 9 +. 6 7 3) . 4-7 9 N 7@'
(76) 37 : 7 $ 9 ' 9 $ ! 8 L" <9 + 7 " M ' $ G = F S(j, k) 9 ' 9*9 7 ' $ 9 " : M ' M=7 >3' $ 9 & +9 ("<$ + $ & 7 M ' 81K ' !" M 759 ' 9*9 7 : 7 2& Γ1 ' : 79 3 : 7 7< $ " 83M 7 $ '3G 9 ) \7<M$ & ' (;M ' 8K (. 9 Γ28 $ Cj 5 Cj+1 Cj+2 j − 1, k − 2} M ' $ 9 ' 9 ' 8K ' :7 ' 3 " :"' 8K M ' 3 " Z: k \ {k 7 8@ 9 7 : 7( M"' $
(77) % ' M7 >3' $ 9 & +<9 (" =9 + 37 ΓM 2' & $ @9 ' 99 7 Cj ' $ 9 Γ" :1 & 37 : 7 $ 9 ' C9 $ j+1 ! 83L 0C" j+2 G \ M0 $ + $ & 7 M 0 ' 8K 0 M ' ) 8 L9 7 1" & & " (. 8 L/: " 37 : 29 & 7 M 0 Γ1 Γ2 ! " : i ∈ Z \ {j, j + 1, j + 2} C M Γ0 & ' 3 " : 5' 8K " 8& +( C M Γ & ' 3 " : i i 2 5 2 k.
(78) OCN TNF[ _ UFR [ S ?ZN.c]
(79) ^U.O W N.[ O[ O W]L% ! !. ' 8K. Cj. Cj+1. l(Γ01 ) = l(Γ1 ). . '3" : ' : 7 0 ' 3 " :"' 8K M Γ2 & Cj+2 Γ01 & 5 ' 8K 0. 8 . . . l(Γ2 ) = l(Γ2 ) $. " 4 ; 5 ./ 3 1 1+- << < ! . P roof. = = 3 t z . x x - . . . = Cj Γ01 Γ1 xj−1 xj tj yj yj+1 ! ) 8 . C . C Γ " 4 - ! . C j! j 0 FBj 1 j+1 " M j+1 ! ! # . . /j+2 . - # ) "C j+1 = 2 0 yj+2 1 1 0 < C j+2 . .3 . Γ5 3 y j+1 5. M tj zj xj xj+1 tj+1 yj+1 zj+1 zj+2 )
(80) .B " . !. %. 1 0 tj+2 yj+2 R1 = x0j−1 xj tj yj yj+1 yj+2 yj+3 Γ1 . @ = 5 < . !. /. 1 ) 0 0 R2 = zj−1 zj zj+1 tj+1 xj+1 xj+2 tj+2 zj+2 zj+3 Γ2. 8 5 5 1 6 6 1*?- ! ). "1 2 . 3 * . 1 < . " tj zj xj xj+1 . 1 . tj+1 yj+1 zj+1 zj+2 tj+2 yj+2 M xj tj zj zj+1 tj+1 xj+1 . )+- 0 - ! . B- 1 < . )Y
(81) . yj+1 yj+2 " ! tj+2 1 zj+2 6 M @ # 0 . 0 R1 Γ1 R1 = x0j−1 xj xj+1 xj+2 tj+2 yj+2 yj+3 < ! 1 5 / # 0 ) 0 0 PB" ./R . 22 " Γ1 2 = .3 . . R22=. z2 j−1 # z j t j yj.y j+1 < t j+10zj+1 . zj+20 z j+3 ! 2 ?; ! ! )YA < Γ1 < Γ 2 <1 . l(Γ01 ) = l(Γ1 ) l(Γ02 ) = l(Γ2 ) " 4 # 0 " < 4 ! 1- . . #< 1 i ∈ Zk \ {j, j + 1, j + 2} Ci Γ02 " < 4 # " < 4 ! ) 0 Γ 1. 0 " < 4 Ci Γ2 0 Γ1 Cj Cj+1 Γ2 . 0 " 4 ) +- =! 5- 1 . Cj+2. Γ1. xj. y. yj. y. j+2. t j+1. zj. z j+1. t j+2. Cj+2. Ý. ¥. x j+2. x j+1. y j. z j+2. Cj+1. D . xj. j+2. y. j+1. tj. Cj. x. x j+1. y. j+1. j+2. tj. t j+1. t j+2. zj. z j+1. z j+2. C j+1. Cj. M ^ ^ ¤ . C j+2. ¥.
(82) ., B .,+- << <2 < ,/& " $ ' &9 : ' 8 M " : 0 & ' 9 " 8" =9 + 7 3). -7 9 N 76'37 : 7 $ 9 ' 9 $ ! 83L/" 9 + 7 " M ' $ "9 ' 9 M 9 7 ' $ 9 " : ' M"7 >3' $ 9 & +<9 ("$ + $ & 7 M ' 8K 1 MG' $ = F9 S(j, ' 9 k) \M ' 8K 2 & MG' M L : 7 ' 9' M(" M M N & 7 7 8%9 Γ7 1: 7@7 > ΓM 9 2M' 9 " M 9/" l(Γ 87 1 ) ≤ l(Γ ' 3 "2 ): $ % l(Γ2 ) Γ1 & $ &' ( $.
(83) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL . 8 ! 1 * 6 Q 5 1; .3 3 .- 6 F3 2 8 . 6< F3 "! @ 3#+- <<2 = = 25 5 5. l(Γ2 ) 26 ! " 4 6 26 . . ! )
(84) . . 2 Ci ! < 0 . ! 6 25 Γ1 C" i+h+1 4 ) 8 . h ≥ 1 = F3 "! ;h3#B+- 0 <. (C ! "i+1 - , . . . , C "i+h =) Γ2)
(85) .B3#B l(Γ #3 . 2 ) h 2 . ! 6 3 . 1 . 1 #3 +-
(86) ! 26 . 0 r = b h2 c 3 1 * . "1 (r) ! < (r) * F # 2" # P roof. " 4 5 26 ) Γ1. M. . . 2. G\M. 2 . . ! (r) (r) l(Γ1 ) = l(Γ1 ) l(Γ2 ) = l(Γ2 ) Ci+2b h c . (r) " < 4 ) 8 . 5 3 F " . ! -2 < , . . ! 2# (r) Ci+h+1 Γ1 l(Γ2 ) +-
(87) 5 . 3#<+-
(88) = ;. .<. 3 3= .3 3 .-) h 0. (r) Γ1. (r) Γ2. 4 , PB < ! ! ! " - ! =- 1 =< . 3 " 1 #3 . G= 0 ). F S(j, k). 6-7 9 N 7' 7 : 7 $ 9 ' 9 $ ! 8 L " 9 + 7 " (. 9 7 8/9 7 M ' $ 9 " : ' M7 > ' $ 9 & +2" 87<$ 2+ $ & 7
(89) "G "7=) 7 F8/S(j, & 7 83L k) 9 ( k ≥ 4) $#% l≥k ' 8K'M 7 9-" $ + $ & 27 & M " =& 7 83G L 9 \ M( 7 : 7 (. 9 k p 6 l + 6p = 4k ( 0 ≤ p ≤ 2 )$ +- -<@ 1 " . B 16 #3 . )/E#B+- << . ! "- 1 M 25 6 5 .-)5+- . ! / 52 G = 2 3 = 1 2 . k . )0
(90) < i.Z k 3 3 = 1 - 2 . . .M Ci−1 Ci M Ci Ci+1 2= 3 3 * 1 2 . . )-P, # . * ) M Ci+1 i+2 ? 25 . C 5 3# 5! . C != 3 = 1 0 ≤ i < k −1 . 2, i + 4, . . .} 0 .j .∈ {i, i + F, . . , e. j )B+- ! < . <G3# \ M 0 Cj−1 Cj A )*PB5. = ) {ei , ei+2 , ei+4 , . . .} | A |= k2 X5 ! <2 ! 16 3 . #, 2/ 3 3 1 - 2 . M . = . . 26 . . 5 . C i−1 F yi zi 0 1 C . i 0 ! )*
(91) -2<0 < #C.i 2 3 ! ) xi P roof.. ei. . . Ci. ' M 7( )"
(92) . 5 . 1 " 2 3 = 1 - 2 . Ci+1 !) . 0 . Ci+1 " . F 1 M . Ci+2 xi+1 yi+1 zi+1 0 ei+2 Ci+1 A . 6 6 " 1 . . 5 # 5 1 . G\M 26 5 = 3 32 1 . . . ) 6 xi xi+1 ti+1. yi+1 yi ti Ci+2 ! yi+1. zi zi+1. G\M. ei. ei+2. ' M 7- );
(93) . - . 1 25 3 ; 1 2 . . 0 C i+1 ! . # . )
(94) M . C i+1 # 0 xi+1 zi+1 xi+1 yi+1 = . z i+1 F@ 1 .0 ) 0 ei+2 Ci+1.
(95) OCN TNF[ _ UFR [ S ?ZN.c]
(96) ^U.O W N.[ O[ O W]L% A .Y / 3 . . . . / . . .. 3# @ . G\M e 0 i+2 !) # zi zi+1 ti+1 yi+1 yi ti xi xi+1 zi zi+1 ti+1 xi+1 xi ti yi yi+1
(97) 6 < . . " .- . 1 3 . = .-) {e , e } {e , e } ei.
(98) .- 25 6 = . . .i+2 "! . i+4 C != # i+4 i+6 1 26 A # 6 @ 2")
(99) ! 5 . 1 6 6 )@ Γ 5 G \ M k . .3 ! 61 . . l = Γ = 3 3 . ! 61 . ej l . A . =< = 1 z i z. i+1 ! 51 . )1+- < y i t i xi xi+1 k ! 5 6 7 # l !l = ! =
(100) < zi-z i+1 ti+1 y i+1 p 0≤p≤ G\M .3 . # 5 1* . ) 8 . " - 2 k < = . p 6 2 −p 1 ) k k Γ. l=. + p + 7( 2 − p) = 4k − 6p. , 5A 1 k " 0 ., .,3#,+- T - @ . . F S(j, k) 6< ! " 3 " ! = -) . " 1 )
(101) 6 # 0. 2. 2. F S(j, k). 3. - DLMZLSWaCNWS ][ O ?N.O` YFNF[ _ WU.O[ N.O 4IS _ L6 ?U. F (j, k). 64 , 2 .
(102) 6-7 9 M N 7=' 7 : 7 $ 9 ' 9 $ ! 83L
(103) " 69 + 37 1 " G = F S(j, k) $ %. 7 8 9 7 ' $ 9 " : M ', ' & 9 " 8 ' 8,$ + $ & 77 >3$ 7 9 " : " K K/' 8K < ' 8 K 2& G\M k j = 25 ": 7)7B 8 ' 8K ": k. j=1. 3$. 8 ! * 6 F3 *5 1 *< . . 1 #3. 1. ! . P roof. 6. =< < . .B # )6E#@+- <<TM . /+- 12 <G " G\M * 5 1- F # 2" # . 26 . * 6 1 0 5G \ M #@ )P ! 6 6 1; . Γ#@1 2= ! Γ 2- 5 ) k l(Γ1 ) ≤ l(Γ2 ) X5 ! << < . " 1 * . 1 #3-/6 1 ! < " 1 0 = - , 1 2@ # 5- ., G .B .B ! M < "- )%E# @2 2#, < G \ M . 16 . # +- # < = F3 6 6 5 . " < 4 5Γ2 2") Γ1. ' M 7/ )
(104) = F3 6 . " 4 5 2=) Γ1 P ! 6 6 1; . # ! - 5 . 3# . C0 {y0 , t0 , . 0 - . 6 ) 8 . = 1 5 # "Γ 1 2 " 0 x0 } yk−1 y0 Γ1 i 6= 0 C i Γ2 < 4 . . 5 2 B< . 5 0 Γ1 y0 , t0 , x0 , x1 , x2 , . . . , xk−1 ) 0 xk−1 = y0 •. A1. k = 2r + 1. 2 . r≥1. .. Γ2. .3 . 6 5 . z0 z1 t1 y1 y2 t2 z2 . . . z2r−1 t2r−1 y2r−1 y2r t2r z2r ..
(105) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL .
(106) 3! {ti }} •. A1. = 3 3 5 1 )
(107) 6< . i=k−1 G .y0 3x!k−1 6 x20<yk−1 # z 03zk−1 6 < # . ) ∪i=0 {Ci \ j=2. k = 2r + 2. 2 . r≥1. .. Γ2. G = F S(2, k). .3 . 6 5 . z0 z1 t1 y1 y2 t2 z2 . . . z2r−1 t2r−1 y2r−1 y2r t2r z2r z2r+1 t2r+1 y2r+1 ..
(108) 3! . . < 3 )<
(109) = . = i=k−1 z0 yk−1 . 3! 5 ." # 6 < # . ) ∪i=0 {Ci \ x0 zk−1 y0 xk−1. {ti }}. j=1. ' M 763)
(110) 5 . " < 4 5 2") . 3 Γ 1 )*
(111) .-. 8 ! 5 •. A1. x0. k = 2r + 1. Γ1. 2 . r≥1. .. Γ1. Γ2. G = F S(1, k). .3 . . x0 , x1 , . . . , xk−1. ). .3 . 6 5 . y0 t0 z0 z1 t1 y1 y2 . . . z2r−1 t2r−1 y2r−1 y2r t2r z2r ..
(112) 3! . 3 * 1 . " i=k−1 x0 xk−1 y0 zk−1 z0 yk−1 . 3! 6 2< # 6 < # . G ) ∪i=0 {Ci \{ti }} j=2. •. A1. k = 2r + 2. 2 . r≥1. .. Γ2. G = F S(2, k). .3 . 6 5 . y0 t0 z0 z1 t1 y1 y2 . . . y2r t2r z2r z2r+1 t2r+1 y2r+1 ..
(113) 3! {ti }}. . < 3 )<
(114) = . = i=k−1 ∪i=0 {Ci \ k−1 .x 30 !xk−1 6 y 0 y=k−1 # 3 z 0 z6 < # . ) j=3. G = F S(3, k). # "X ! < , @- ' $ 9 " :, ' & 9 " 8 ' 8%S 6 U 1 # " 1 1 *5 < . G . # * C ! 2 & .3 # 11 * # 1 <0 2 . 1 G " 1 6" 0 . .B # ! ) 0 M. G. 2. G\M. E#,
(115) 1 < . #%3 . 1 < . #% . . j ∈ {1, 3} - . @1 = #@ 1 6k ≥ 3. 1* #3," . 0 " k1 j=2 F S(j, k) 2 ! # =0 < . ., # )<EM#/+- << F S(j, k) \ M 1; #3=< ! @ " F1 S(2, k) k ≥ 4 " 1 . . 05 < . ./ # M 6< #@ .3 . 5 # 6 1; 2. ! ) F S(2, k) \ M 6
(116) .@2= = =1 26 . ) 0 L : ' G = F S(j, k) M 2 & ' $ 9 " /: ' & 9 " 68 ' 2 8 "' 8, K " 8& +, M " K K' 8K ":. . k. j=1. 3$.
(117) OCN TNF[ _ UFR [ S ?ZN.c]
(118) ^U.O W N.[ O[ O W]L% . PB. = @V
(119) F,O6 ;V< 1 5 .- 8 #3< ! . F S(1, 3)
(120) < 5S U );P, 5 - F < .= Q3. 2. ! ! " 1 < " 2 . .@ ! 5 . 8 . ) ) . % 75$ 3: " ' 8K M " K K j=2 k $. ' 9 $ 8K 7 > " 6'5L : ' . G = F S(j, k). M. 4. 6' 8K " 8 & +. PY . . -3 . # " 1 -"! ; ; - 25 # j=2 #B
(121) 3)
(122) .B+- k<< 2 = " .3# " 1 0 " = - B 1 2 2<3 # )95 . 6 < = . F ) P . G . 53 3#@
(123) 4 6 . .-) j=1 3 k F S(j, k) 6 .@ . " 1 ) A 1 6 .@ .@ #+- << 5T3 < . 0 0 k F S(j, k) 2 P roof.. . 4 , . X6 - . 5 1; = 3 ! = .5< # Q2 . 1 - 1 6< " . ! ) .@2 < . / ! . = 1 . 3 = 1* #3- . < . 3 " 12 #3 F S(j, - k) k < 0 .3 # " 1 " 1*26 / 5 < . .B #" X6.@ < 2" . 2 5 -5 2 .-) +- ! . 0 " 1 < . 0 1- #3 . 26 ) 2 F S(j, 2p) 8 ! * 6 .= 3 3 * 1 - M 2 . . 1 .3# p ≥ 2! C2i−1 C2i ≥1 ! 6<< . @ 1* M #3- 5 6 < # " I= W . ./ ) . i 5 2 2.0 ! . ! " M 25 . )
(124) " 5 " 0 C2i C2i+1 0 ≤ i ≤ p − 1 ' M 7 F ! ) .!) {y2i y2i+1 , z2i z2i+1 } ⊂ M. ' M 7 # ! ). {x2i x2i+1 , z2i z2i+1 } ⊂ M. 'M7. !). {x2i x2i+1 , y2i y2i+1 } ⊂ M. M ∩ (C2i ∪ C2i+1 ) = {x2i t2i , x2i+1 t2i+1 }. . .-. M ∩ (C2i ∪ C2i+1 ) = {y2i t2i , y2i+1 t2i+1 }. M ∩ (C2i ∪ C2i+1 ) = {z2i t2i , z2i+1 t2i+1 }. !) !).
(125) " ! . 3! @ . 6 @N & " $ )A . F !F " 2i+1 0 # #
(126) # ! ! ! C2i ∪ 3C QB ,,N & " $ " <9 + 7 0N & " $ " 69 + 7 X N & " $
(127) " 69 + 37 ! )*
(128) . 26 5- 1 < . 1 #3 Y.5 - .< *5 ZC ! . 1 F S(j, 3 Q 2p) ; # . . )*A . M 2.0 2 - 1 6< . 1; #3 p . 6 .3 #@ # @2 % 16 . ., M - < 216 F S(j, 2p) )%
(129) 3 Q p. {X, Y, Z}.
(130) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL . " 86 9 ' &*N & " $ < . / 3 Q = %9 7 : 8' & 2p−1 ∪ C2p N &" $ )
(131) = F3 < - 3 Q 6 5. 5 . C = 6 . ! 5 3 Q ) 0 E#=+- <<( 3 . " # - 1 . " . #= 1 6! . C ! .< # F )*A S(j, ; ; 2p)\M .< #=< ;6 6 ; 2= .F S(j, ! 2p)\M 3 Q 0 . = 1 = #3-0 )=
(132) . <. 5 . 30 ! " # " 1 . / . #@ 1 #/ " 6 . W ! . <1 < 2B . ! 3 Q XY XZ Y X . )
(133) -< < . " " 0 # = . 25 5 Y Z ZX ZY 6 C0 ∪ C2p−1 ."1 = # "1 .B F3 < * . W ! . 0 )5X6. 0 F3 < . W ! j.<∈ {1, 2,3} . #62 . 25 ; . {X, Y, Z} ! 6 1 . 6 #35 1- 6 . 3 Q . < C0 ∪ C 1 36 5 . 5 #3= 1; 5 . - 3 Q );P,5 ! - C2p−1 ∪ C2p # / 3 / F3 = 3 Q / . . 1 . -1 3#/ 3 j = 1 . x2p−1 z0 y2p−1 x0 z2p−1 y0 j =2 x2p−1 x0 y2p−1 z0 . 1 #< 5 3 3 . )*
(134) .-3 z 2p−1 #<y 0 z2p−1 z0 2p−1 x0 y2p−1 y0 1 # j = 23= 5 51 x 2 . < ! ) 0 0 C0 ∪ C 1. ,-7 9 N 7,'2 7 : 7 $ 9 ' 9 $ ! 8 L " /9 + 37 " M 2.0 (. 9 M ' $ @ 9 ' 99 7 ' $ 9 " : M'
(135) ' & 9 " 86 ' 8/$G+ $ = & 7 F S(j, 7 8B2p) 9 7 $#% (" : N K pK 7 ≥8B2) 7 > 9 : 7 ' &;$ " 8 *-L '3: 2' & 9 " 8 M' : G7 \ M XY 5 Y Z. ' 8K. XX 5 Y Z XX 5 Y Y. ' 8K ' 8K ' 8K. ZX ZY ZZ. ": ": ":. F S(1, 2p), F S(2, 2p), F S(3, 2p) $.
(136) 3! 6 . #%- 1 < . 15 #3- 1 ! " 1 <@ . M .Y # 2.0 #% F S(j, 2p) #%/2 %2 1 . G \."M * . 6. 26 . . ! * . ! p 6 = ! " 2 {X, /SY, . Z} - 0 U S < . - U- . 6"1 0 . . W ! .-)
(137) .-326 6 .<- . " . 6. ! "- 1- ! 1 0 # µ0 (j, 2p) < . 3 . )+- *! . 3# 0 F S(j, 2p) µ2.0 (j, 2p) ! 0 2.1 # #3 . 3 .. ! " = 1 1 < . . . 5 1 3 # µ02 (j, 2p) 2.0 #3- ! < .3 # B/ < . .Y # . ) * # 2.1 2 F S(j, 2p) . ) 0 0 0 0 0 µ2 (j, 2p) = µ2.0 (j, 2p) + µ2.1 (j, 2p).
(138) . µ2.0 (j, 2p) = µ2.1 (j, 2p). 6 7=8' N 7 : M 0 " 7 : 7 $ 9 ' 9 $ ! 83L M(" 59 + 7 $ " & 7 % & µ2 (j, 2p) 2 8 N+ 0 7 89 ' : +<9 " ' & 9 " 8 ' 8/$ + $ & 7 M 8 F S(j, 2p) (j ∈ {1, 2, 3}) ' : 7=L ) 7 B µ02 (1, 2p) = 2p+1 + (−1)p+1 2,.
(139) OCN TNF[ _ UFR [ S ?ZN.c]
(140) ^U.O W N.[ O[ O W]L% ' 8K. . µ02 (2, 2p) = 2p+1 , µ02 (3, 2p) = 2p+1 + (−1)p 4 $. . * ! # 1 *< . 3 1 #3-. )*+- . 0. . . α = 2< 6 . . 6. #@ 3 . ! )6+- 2.0 β1 p -5 " 5 2 = 1 . {X, . Y, Z} . .< 2@ . ! A αβ .3 * 0 p {X, Y, Z} 0 . @ . 3 . #@< = 3 . 51 )6I= . = =. ! "- . . . # 12 ; . pα # p )*+- p 6 * 12 β1 . . Aαβ . .= 2 " . a αβ ! 6 B .3αβ 3 . . . " # .p < . 3{X, . "Y, # Z}) 0 0 α β I5 . =3# p 5.3! " = 1;2 6 . p ) * # b5αβ @.3! " @ 1"2 1= B . αβ . %.B 2 . ! p 0 .3 = . @- . . . @ # )
(141) . p )
(142) 0 α 2p−1 aαβ + bpαβ = 2p−1 P roof.. . 1 1=2 . p =2 . p−1 . 6 3 . 0 β B A 1 = 35 1*2 . p−1 αβ 62 . p αβ ) 0 β Aαβ Bαβ . 1 5 # )P,=. =
(143) 3! p bαβ = ap−1 0 p ≥ 3 apαβ = 2p−1 − ap−1 αβ αβ 1 . 2 1 )@A 1 2< @ < 0 0 a2αβ = 2 α = β aαβ = 1 α 6= β α = β ! .3 C ! . . 1 )A 1 2 u2 = 2 up = 2p−1 − up−1 p≥3 α 6= β 5 6 6 ! . 6 C ! . . 1 ) v2 = 1 vp = 2p−1 − vp−1 p≥3 . 1 )
(144) 0 .@2" 0 . 2 p−1 p vp = 31 (2p + (−1)p+1 ) p≥2 E#+- <up = 3 (2 + (−1) ) µ02.0 (1, 2p) = apXY + apY Z + apZX = 3vp = 2p + (−1)p+1. . . 8 . . µ02.0 (2, 2p) = apXX + apY Z + apZY = up + 2vp = 2p µ02.0 (3, 2p) = apXX + apY Y + apZZ = 3up = 2p + (−1)p 2 µ02 (j, 2p) = µ02.0 (j, 2p) + µ02.1 (j, 2p). .@ = . . ! . ! ). , *PB" = . . ! "- µ2 (j, 2p) = 2 × 3p Q32 . 8 ./ ! ). . . ). µ02.0 (j, 2p) = µ02.1 (j, 2p). 2. . 6 6 < "2 / 1 1 5 . " 1 #3 .. µ02 (j, 2p) ' 2p+1. 2. F S(j, 2p). 6 . ? =< ./ M 9 : " 83L ' 9 $ ! 83L ( 8K ' $ 7 K ' 9 $ ! 83L ! 5 = (V, E) < . 0 ! B G 5 . 2 3 = 1 54 . @3#/ .B 3 3 1 = )
(145) S S G 6 " = 1 3 3 5 1* " ! B 1 . 3! @3# ) PB 6 S. G. V (S).
(146) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL. T. . " ! = . @ 1 =< . @2 B 5 "! . ., 1 2 . < . 6 62=0 3' 7 L37 : M=L : ' . =S U 5 = ! 5 "7 '! 9 ' N & 7 ! )P,= ' 7 L 7 : M ' 9 $ ! 8 L"" 1 . 1 ! " 2 / 6 =! . ./ 1* 2 . < . . M G ) 8 ! ! . 6 M B ! ) MR ) X6. 3 35 B1 = V (M B ) > 7 K6 1- 0 . R =5 3V (M .R ! ) 08 . 6 G 6 1; F 3 3 0 " 1 0 . . ) E(G) \ M 2 G\M | B |=| M | 0 6< ! " 3 3= ! = . 1 6 . #
(147) 3! # ! 6 G 3 3 ! 0 . ) 8 31 . . S U- . = MB ∪ MR | MB |=| MR | S U-1 5 5 - M 5 1; 5 ) A . 6 ! .2= < .= ! 6 1 . 1 26 @" 0 j k 6 3 ! -) F S(j, k). . ∪ MR. M'
(148) 3' 7 L37 : ML : ' 2 ' 8K k ≥ 3) M"' 3' 7 3L 7 : M ' 9 $ ! 83L," 9 7 8 M '/37 : 7 $ 9 ' 9 $ ! 83L," M =9 + = 7 MB G, M 1$ G = F S(j, k). 8 ! - . * 1 #36" . < ! 2 ! 1 3 . # " < 2/ 3 3 " 1 2 . . ;1 " . . . M C0 C1 )
(149) . . ) 8 ! 5 x0 x1 . y0 y1 = {t1 z1 } 8 .{t 0 z0 } 6C 1 ∩ . M << . C 0 ∩ M) = . . -x 0 x 1. y0 y1 MB MB t0 z0 t1 z1 )"
(150) 5 5 - "- ! = . < . ) M \ M B = MR R E#< #3<< #< 5 6.= 2 3 3 1 M 2 . . )*
(151) . MR 1 " .< 3 3< 1 2 . . 1 = . C 0. ; C. B .< 3 < 1 MB C0 C1 x0 x1 - 2 . . 1 " . . ) 8 . . . MR < . 3 C 0= .<C 1 3 3y=0 y1 1 . < .3 M B 3 .-M )
(152) R ! 6 M C0 ∪ C1 M 1 6< . 1; #3- ) P roof.. 6 (j = 1. ' 8K. M. M<' 3' 7 L37 : M"L : ' . 2 . G = F S(j, k) k ≥ 3), ": ": ' 8K k≡1 2 (mod 3)) (j = 3 k ≡ 0 (mod 3)) $. +- . - ! < . % 1. 9 7 8B7 9 7 :. )YE#%+- <<. P roof. MR 1 M "=M B ∪. B 16 #3- ) 8 ! <2 ! " G 16 3 . # M ) 8 . 1 B . B< . % <. 3 M 1 B ∩ E(C - 20 ) = . {x0 t.0 } ) 8 ! M-B ;2 ! " " 1 . # * MB 3 < . 4 . . C 0 C1 ) . 25 . MR C0 C1 y0 y1 C0 C1 ) 8 . . . @ . *2 .% " < C2 MB 1 . MR W F 5 "- . " 1 < " 3 " 1 . x0 t0 ∈) M /B " y 0 y 1 # ∈ MR . )
(153) M B MR {t1 z1 , y2 t2 } ⊂ MB {x1 x2 , z2 z30 } ⊂ MR. ! . C !" . W ! ./ 6 ./?- ! ).
(154) OCN TNF[ _ UFR [ S ?ZN.c]
(155) ^U.O W N.[ O[ O W]L% x0. y1. y0. x2. y. 2. t t0. t1. z0. z1. C. 0. D . x1. ¾ ¼ DF ¥. C. 2. z2. C. 1. MB. . ¡. 2. ^ ¢ . MR. ¡. F ¢ ¥. A1 .%2 < . ) 8 0 k ≥ 4 3 y4 ∈ M R " 3 ; ! ./ . 6 z 2z3 ∈ 5 M <R x63 t. 3 ∈ M.B 2 y./ = 5 C3 C0 <@! . C ! 3 ! < . ! % . M = M B ∪ MR x0 t0 ∈ M B .B )PB 1 #/ . < 1 y0 y1 ∈ M R 0 . . . . <- 2 .Y @F S(j, 26 k) . 0 )YPB Ck−1 C0 . *5 . 5 . = . ! " . " 1 M B * =. 3M ! ∩ (∪i=k−1 E(C )) M 2 R # i=0i=k−1 i ) ∪i=0. {V (Ci ) \ {ti }}. ' M 7/ ) 26 ) k = 3p p≥1 PBB . 0 0 x0 t0 ∈ M B yk−1 tk−1 ∈ MB xk−2 xk−1 ∈ MR ! < )
(156). ! .z k−1 z0 = zk−1 z0 ∈ MR zk−1 z0 yk−1 y0 xk−1 x0 zk−1 z00 ∈ MR = 3 3 6 1 . 25"! 6 ) 0 j=3 F S(j, 3p) ' M 763) 26 ) k = 3p + 1 p≥1 6 x x ∈/ E(G)! z z ∈ PB" k−1 0 k−1 ! )Y
(157) 3!k−2 .0 x00 t0 ∈ MB xk−10 tk−1 ∈ M B MR zk−1 z0 = yk−1 y0 ∈ MR yk−1 z0 ∈ MR yk−1 z0 . = 3 5 1 . 2="! 6 ) xk−1 y0 zk−1 x0 F S(j, 3p + 1) 0 j=1 ' M 7 3) 26 ) k = 3p + 2 p≥1 PB, . 0 0 x0 t0 ∈ M<B zk−1 tk−1 ∈ M! )% B yk−2 yk−1 ∈ MR
(158). ! .z k−1 z0 = xk−1 x00 ∈ MR xk−1 z0 ∈ MR xk−1 z0 yk−1 x0 zk−1 y0 = 3 3 6 1 . 25"! 6 ) 0 F S(j, 3p + 2) j=1 , A 1 25 <1 +- <<% B 1 0 # k ≥ 3 =. " ! = -)<
(159) = = ! =26 . =3 - . 0 k RF S(2, 2 =k) . Q " = . F ) 4 0
(160) .-2= .@ =1 26 . ).
(161) K P Q.TIU.RV.RLMWIXIY\H]R[ _ _[ LMZCN.O` K P abN.O]LZ cL. ! .
(162) . 6 1 " : ' 8K 9 7<L : ' M@' j ∈ {1, 2, 3} k ≥ 25 G = F S(j, k) 3' 7 L37 : M=L : ' 2 "' 8K, " 8& +
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