Circular Choosability
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Circular choosability Frédéric Havet — Ross J. Kang — Tobias Müller — Jean-Sébastien Sereni. N° 5957 Juillet 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--5957--FR+ENG. Thème COM.
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(29) joP
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(52) f|jbdNzghf{joP0;j %bP
(53) OOP
(54) e!_Wf|e!_ajoc^r^dNf|_aPQqhj V)c0Kh0j5 P
(55) _;K,d {%u% e qNjoP
(56) Of|P jz{f|P d§3eNc0d] j
(57) q;c0er^c#e]e
(58) #h_af{¢0P OQP e!_ ªdNeNP£!dNP ]`_af{c0e¬r^P .K!dqNjoc0q;c#] r^P%] bz{f{£!dNP%] bf{job
(59) dNz{h0f|joP ] e]adNf_oP0eNc0d] r
(60) Oc0e!_ojac#e]£#dP0Gq;c0dj_ac0dN_0jhqNKNP G "i dNfg] NeNc0d]
(61) _adrNf|c#e]dNeNP e
(62) jhz{f{]oh_of|c#eBr^PWz{hQbKNc#f{]af{]o]ohNf{z|f|_ cch(G) = O (ch(G) + ln |V (G)|) b
(63) f|jbdz{h0f|joP0 z|c#jo]o£!dNP"z{h_hf{z|z{Pkr^P zghLz{f{]`_aPkr dNe]ac0OOP_§P%]m_G¨c0eb_af{c0er^P ]ac0e r^P 0 j 0 c0d]bc0e]`fgr
(64) joc0e] #h0z|P OQP e!_nz|P eNc#ONjoPr^PbKc0f|¦bf{jbdNzghf{jaP r^P%]U0jhqKNP ]LqNzghehf{jaP%]
(65) Vªc0Kh0jnh5r^P Oh0er zgh¢hz{P
(66) dNj rNP sup{cch(G) : G P%]m_nqNzghehf{jaP } c0d]qNjac#dN¢0c#e] £!dNP 6 ≤ τ ≤ 8 "bP5£!dNfk¨c0dNjoeNf|_dNeNP j
(67) q;τc0e:= ]`Pe
(68) #h_af{¢0 P 5dNeNPSh0d^_ajoPS£!dNP%]m_of|c#e)r^PSV)c0Kh0j ke e§.eNc#d
(69) ]
(70) _adrNf|c#e]>z{Pec0OjaPrNPSbKNc#f¦ b
(71) f|jbdz{h0f|joPUr^P ]3#joh0qNKNP%]"qNz{h0ehf{joP ]"P_qNz{h0ehf{joP ]"P¦, _
(72) jof|P dNjo]3rNPnOhf{z|z{Pnr^c0ee
(73) P#0P_bP
(74) z{dNfrNP ]3#joh0qNKNP ] rNPrNP
(75) e]af _ c#jae
(76) P0 ¶ % ´ D bKc0fg]`fg]a]ohf|z{f _ °bf{jbdNzghf{jaP#neNc#ONjoP¯r^PbKNc#f¦ bf{jbdNzghf{jaP#nbc#z|c#joh_af{c0e§U#joh0qNKNP ] qz{h0ehf{jaP%]
(77) ,#joh0qNKNP ]qNzghehf{jaP%]P
(78) ¦,_
(79) jof{P
(80) dNj]
(81) NrNP
(82) e]af _ c#jae P.
(83) .
(84) . G""! k %
(85) &')( c0 fg]Uh¨deb«_of|c#e #GP_ PhQ0jhqK§ her hjoP~_mcQf|e!_aP 0P jo] Nh c ¨joc0GO =V (V,_ac E){0, . . . , p − 1} ]adbpK¯_aKh_ q q ≤ |c(u) − c(v)| ≤ (p,p −q)q$ ¨c0j he,\ªP rN0P Guv ∈ E QJLKP b
(86) f|jbdz{h0jLbKjac#Oh_afgb>e,dNOSP jUc_aKNP #joh0qNK G fg] h#r^Of_]Uh (p, q)© bc0z{c0djaf{eN . p χ (G) = inf :G ¡c0jkh0e,\Sf{e!_aP
(87) #P
(88) j a !P>rNP
(89) eNc0_aPn,\ [a] _aqKP>f{e#_oP
(90) jo¢hz [a − q + 1, a + q − 1] "pUP
(91) Ohjo®_oKh_f uv ∈ E _oKNP
(92) e c(u) = a ⇒ c(v) ∈/ [a] NUKNP jaPW_aKPbc0OqNdN_oh_of|c#e]UhjoPOc,rNdNz|c p * +,&- &(.'/"0'1 c f{]hSOh0qNqNf{eNS_aKh_Lh#]a]af|#e]3_ocSP ¢0P ja\¢0P j`_oP¦c hS]`P
(93) _c0Gf{e#_oP
(94) #P
(95) j]
(96) b hz{z|P%r
(97) % * e L L$ %
(98) G&'2( c G f{]hxOh0qNqNf{eN c _aKh_Sh0]o]`f{0e]W_acªP
(99) ¢#P
(100) jov\)¢0P j`_oP¦ v ∈ V h b
(101) c0z{c0dNj c(v) ∈ L(v) * t $ (p, q) $ +,& $ 3%&( '1/40' L fg]kh z{f{]`_h#]a]af|#eNOP
(102) e!_k]adbK_aKh_ #¨c0jhe,\¢#P
(103) ja_aP¦ h0er v ∈ V L(v) ⊆ {0, . . . , p − 1} %≥
(104) tq .
(105) 0 f_oKNP
(106) joPnP
(107) ¦,fg]`_o]kh % 5 &')( !f' P0 f{] JLKNP~0jhqNK G fg]k]ohfgrQ_ac;P (p, q)|L(v)| ;c_oKh (p, q)© b
(108) c0z{c0dNjof{eNWh0erhe L© bc0z{$ c0Ld$ jaf{eN¡Nc0j h0e,\jaP%hz^e,dNOSP j t(p,≥q)1 $ Lf|_ $ f{] t $ (p, q) $ c%673 .c
(109) 0 f|f|_Uf{] (p, q)© L© b
(110) c0z{c0dNjhNz{PU¨c#jLh0e,\ t© (p, q)© z{f{]`_ © h0]o]af|#eNOP
(111) e!_ L #h0]`_ G fg]
(112) & + t $ %673 .
(113) 0 f|"f_>fg] t© (p, q)© bKNc,c#]ohNz{P~¨c0jnhe,\ p, q kJLKNP
(114) +,&8%2 /4 9:'1 /4%0<; c0j 5 =%6 90 '1 /4%0?> c fg] G f{]>bf{job
(115) dNz{h0jaz{\ t© bKNc,c#]ohNz{P }. cch(G) = inf{t ≥ 1 : G * ]k]aKNcUef|eª{%u@ ¨c#jkhe,\S0jhqNK G A @\bc#e,¢0P
(116) e!_of|c#e§Pnr^P
(117) eNc0_aP ; c0jr^f{ehjo\ > ch(G) z{fg]m_>bKNjoc0Oh_afgb>e,dNOScch(G) P jU!\ ≥ch(G) −1. c_aP_oKh_U_oKNfg]nr^P eNf_of|c#exc3bf{job
(118) dNz{h0jUbKNc,c#]ohNf{z|f|_m\Br^
(119) f B.P
(120) j]U]`z{f|#K!_az{\¨jac#O _oKNP c0eNPf{e!_ajoc,rNdbP%r f{e|u%@0f|eSUKNf{bK p f{]"joP £!dNf{jaP%r_ac;PUh_"z|P%h0]`_ 2q ¤¥PKh¢0Pkc#q^_aP%r¨c#j3heNc0_aKNP jr^P eNf|_af{c0e§0P%b
(121) hd]`P _oKNP>joP £!dNf{joP
(122) OP
(123) e!_rNc!P%]kec_]aP
(124) P O _acQOh0®0P>]aP
(125) e]`P# eBqhja_afgbdNzghj%!f_Lbc0e!_ojoh#r^f{b_o-] #§P
(126) OOhQw ck{%uD C der^P
(127) j~_aKPQr^P eNf|_af{c0e¯0f{¢0P
(128) e°f|e |u%@;_oKNPQb
(129) f|jbdz{h0j~bKNc,c#]ohf|z{f_m\ck_aKNP]`f{eN#z|PSP r^#P K c#dNz{r°P c#eNP0Jcx]`P P_oKNf{] G]adNqNq;c#]aP_oKh_ L , L ⊆ {0, . . . , p − 1} hjoP#f|¢#P
(130) e)Uf|_aK |L |, |L | ≥ (1 + )q h0er p ≥ 2q _f{]WeNc0_rNf E5bdz__acx]aP
(131) PS_aKh_~_oKNP
(132) joPSOSd]`_WP
(133) ¦,fg]`_ c ∈ L , c ∈ L ]adbK)_oKh_ q ≤ !]ac_aKh_ c#dNzgrSf{er^P P rKh¢#Pnbf{job
(134) dNzghj bKNc,c#]ohf|z{f_m\c0eNPnder^P
(135) j _aKNPWr^P ef_of|c#e |c ¨joc0O − c||u%≤@T~p −e q_aKP c_aKP
(136) jUKKh0er #GP
(137) OOhwc0{%u%f|OqNz{f|P%]_aKh_ cch(K ) ≥ χ (K ) = 2 Vªc#jaP c¢0P j "¨c#j]ac0OP0jhqNK] G k¨c0j t ]az|f{0K!_oz|\¬z{P ]o]S_aKh__aKNPªbf{jbdNzghjbKNc#f{b
(138) P e,dNOSP j k_aKNP z{f{]`_ h0]o]`f{0eOQP e!_o] ¨c0j UKNfgbK fg] eNc0_ q)© L© b
(139) c0z{c0dNjhz|P]ah_afg] P ]W_aKh_ p fg] eNc_ _oc,c t© (p, q)© zghjo0P b
(140) c0Oq© hjoP rB_ac q ¡c0LjnP
(141) ¦^h0OqNz|P#h#]UGf|ex_oKNPqN(p, joc,c"c3i joc0q;c#]af_of|c#ew ; f|f > c0eNPb heª]aKNc _oKh_ f| t ≥ 2 _oKNP
(142) eBh t© (p, q)© z{f{]`_ © h0]o]af|#eNOP
(143) e!_ L ¨c0jUKNfgbK K f{]eNc0_ (p, q)© L© bc#z|c#dNjoh0Nz{PL]oh_of{] P%]3_aKh_ NUKNP
(144) joP h#] p ≤ 4q ¤¥Ph0z{]acUjaP Ohjo®ncch(K _aKh_G_o)KNP
(145) =joP 3fg]Gh0ehz|_aP
(146) joeh_af{¢0P0bc#e!_af{e!dc0d]§rNP eNf|_af{c0e¨c0jGb
(147) f|jbdNzghjbKNc,c#]ohNf{z|f|_m\0 Rc#e]adNz_ª nc#jª|u%U¨c0j_oKNf{]Br^P ef_of|c#Ge FnKNcP
(148) ¢#P
(149) j% f|e _aKPªjaP%]m_cW_aKPqhq;P
(150) j%PxUf|z{znjoP ]`_ajof{b_ c#dNj]`P z|¢#P ]P
(151) e!_af{jaP z|\5_ac +,& (p, q) $ %
(152) &')( ^_aKPbc0ebP
(153) qN_LP Kh¢#Pklmd]`_nc0dN_az{f|eNP%rKNP
(154) joP0 JLKNP¯qhq;P
(155) jBf{] c0jo#h0eNf{]aP r h0]5¨c0z{z|cn] C°f{e _aKP j]m_]`P%b«_af{c0eGUP°rNP hz~Uf_oK bf{job
(156) dNzghjBb
(157) z|fg£!dNP ] ; i joc0q;c#]af_of|c#eB} > h0erP
(158) ¢#P
(159) e5b
(160) \,b
(161) z|P%] ; Rc0joc0z{z{h0jaI \ H#ikjac#qc!]`f|_af{c0eBu h0erBRc0elmP b«_odNjo<P J > JLKP
(162) e§,f{e_aKNP ]aP b
(163) c0er¥]aP b«_of|c#e§GPbc#e]`fgr^P j_aKNP5£!dNP%]m_of|c#e§G#f|¢#P
(164) eh #joh0qNK G GcLUKNP
(165) _aKNP j cch(G) fg] ;c0dNer^P r Uf|_aKf|eªhbc#e]`_ohe!_LÄh0b_ac#jUc ch(G) FPh0e]`P
(166) jLf{e_aKNP q;c#]af_of|¢#P UKNP
(167) exjoP ]`_ajof{b_aP%r5_oc} © bKc!c!]ah0Nz{P . . c. q. q. 2. 1. 2. 1. 1. 1. 2. 1. ÌÌ ÏKL«÷&LM. 2. 2. 2. 2. 2. 2. 3. 3. c. 2.
(168) 04 '2(.
(169) 0
(170) 00'1 H. # joh0qNK] ; Rc0joc0z{z{h0ja\B0 > #h0erQ_aKNP e5d]aP>h qNjoc0hNf{z|fg]`_afgbL_aKNf{eNeNf{eNQhjo0dNOP
(171) e!_3_occ#dNer_oKNP>b
(172) f|jbdNzghj bKc0fgbPe!dO;P
(173) jch¯#joh0qNK¬,\ h¥bc#e]m_he!_QÄh0b«_oc0jc UKNP jaP fg]Q_aKNPxe,dNOSP j c0U¢0P
(174) ja_afgbP%] ; JLKP
(175) c0joP
(176) O %} > e_aKP5_aKf|jr ]`P%b«_of|c#e§"P5(ch(G) f{e,¢0P%]m_of|!+h_alnPhªn)#P
(177) eNP joh0z|fg]onh_af{c0e¥cnb
(178) f|jbdNzghj bKc!c!]ah0Nf{z|f|_m\0kb
(179) f|jbdz{h0j bKNc,c#]ohf|z{f_m\#3UKNP
(180) e fg]h¯¨dNeb«_of|c#e c>_oKNPªr^P
(181) #jaP P ]her h_`_aP Oq^__ac P%]m_hNz{fg]`K¬h0e¬heh0z|c#0dNPQf¨©c0j_aKNPBjaP%]`dNz|_c0mf{er^fP qP er^P e#_oz|\# @c0joc,rNf|e Hh0er jr#] "pUdNNf{e her Jh\,z|c#jU u ; JLKNP
(182) c#jaP O u > JLKNPn¨c0dNja_aKB]aP b_af{c0e;P
(183) #f|e]kUf_oKBh£!dNP%]m_of|c#ecGVªc#Khjn !UKNch0]a®0P%r ¨c#jL_aKNP ¢h0z|dNPc0 fg]LqNz{h0ehj }. τ := sup{cch(G) : G ¤¥P ]aKh0z|z qjac¢#P_aKh_ 6 ≤ τ ≤ 8 ; JLKNP c0joP
(184) O } H)heri jac#qc!]`f|_af{c0e .} J > ª¤¯PB_aKNP e 0f{¢0P5;c0dNerN] ; i joc0q;c#]af_of|c#ex}!wh0erJLKNP
(185) c#jaP O #u > ¨c#jL_aKNP qhjhOP_oP
(186) j f{]nqNz{h0ehjUh0erKh0]L#f|ja_aK ≥ k}, τ (k) := sup{cch(G) : G h0er.Oc0joP#P
(187) eNP joh0z|z{\0_aKNPb
(188) f|jbdNzghjWbKNc0fgbPSe,dNO;P
(189) jWck#joh0qNK]~Uf_oK°;c0dNer^P r¯r^P
(190) e]af|_m\ ; i joc0q;c#]af © _of|c#e]n}h0er !} > "¡c0jL_oKNPheh0z|c#0c#d]kqhjhOP_oP
(191) j f{]nc0d^_oP
(192) joqNz{h0ehjUh0erBKh0]L0f{ja_aK ≥ k}, τ (k) := sup{cch(G) : G P b
(193) h0e qNjoc¢0PheP¦Nh#b«_UjaP%]`dz_ ; JLKNP
(194) c#jaP O !w > JLKNPB¨c0dNjQOhf{e¬]`P%b«_of|c#e]h0jaP5joP
(195) zgh_of|¢#P
(196) z{\¥]aP
(197) z| © bc0e!_hf{eNP rher Oh\0"Uf_oK ¨P
(198) P¦NbP q^_af{c0e] "P joP h#r5f{er^P qP er^P
(199) e!_oz|\cP h0bKc_oKNP
(200) j% «" U ! U ¤¥P j]m_5Ufg]aK _ac qNjoP ]aP
(201) e!_Bh¥h0]afgb_ac,c0znUKf{bK Uf{z|z~P°b
(202) jadbfghzL_ac¬]aP
(203) ¢#P
(204) jhzUc0Wc#dNjjaP%]`dz_]
(205) e _oKNP5¨c#z|z{cUf{eN°z{P
(206) OOhN G fg]#f|¢#P
(207) e§3f{e!_aP 0P
(208) j] p ≥ q hjoP50f{¢0P e§ t ≥ 1 fg]Sh0z{]ac)#f|¢#P
(209) e§ h#]fg]Sh t© h0jaP z{fg]m_ h0]o]`f{0eOQP e!_ S¡NdNja_aKNP jaOc#jaP#.]ac0OPQ¢0P j`_of{b
(210) P ] qNjaP%bc#z|c#dNjaP%r (p, q)© h0ersP hjoP _oja\,f{eN _acDL P¦,_aP erI_oKNP b
(211) c0z{c0dNjof|e h0b bc0jr^f{ueN , u_acD, .]`.c#.OQ, uP c0jr^P
(212) jo(p,f{eN q)c© L© _oKNP ¢#P
(213) ja_afgbP ] ² ¶ ² µ² ² Ä ² ^ ² ¶ · ² ´ (v = u , . . . , v = u , v , . . . , v %^ '" ! ² Q ^ $ ² #² ³ ¶ &´ % ² ¶ ¶ )G ² · #² $² ' NN ¶ ¶ ² ² (. ¶ µ Ä ) * ² + * · $² ' ²)¶ · ² , ¤¯P]oh\Bhb
(214) c0z{c0dNj a f{e L(v ) fg] 0&- 0'/..
(215) 0 f"_oKNP
(216) joP P¦^fg]m_]n]`c#OP b
(217) c0z{c0dNjof|e c_aKNP5]adNN#joh0qNK¥f|erNdbP%r¯,\ {v , v , . . . , v } ]adbK¥_aKh_ c(v ) = a her c (p, q)© L© joP ]aq;P b«_]_aKNP qjaP%bc0z{cc0djaf{eN 1 1435360 F = {w , . . . , w } ,4 60:0 76:' 0(36% % 76 v 8 &/4
(218) 0"'/..09- ' 0² ¬2 o. 1. 1. 1. k. k. k+1. 2. k. |V (G)|. j. 1. 60:9..0&')($:; 0 6 (i) P > wi. Ei. ..0'. 6 "
(219) 0% 3&. i:xi <2q (2q − xi ). 010 08<k6I0&- 0'/..
(220) 0:%
(221) %=6. 09.- 0'.. 50 %
(222) %?6 40% .% x09i- ≥ i 0'1. .
(223) 0:
(224) %":. j. j. j : wi 1 ≤ i ≤ k. 0'. vj. 6 "
(225) 0% 3&. 0&- 0'/.. .
(226) 0 %
(227) %@6 0% < P xi ∈ {1, .09.-. ,02q '.− .
(228) 1} 04%
(229) % 6 60A6 5 ')( B."C i |L(vj )| − i (2q − xi ) 8 6. = % 0 .. % , . ' . ' 9 0 . ' " / 0 ' 0 E . D i Ei [ai , . . . , ai + xi − 1] F. 6 (ii) > /4w&i. 6 /4&. 2. 60'. |L(vj )| − vj. 6 . ë¨î̧ëÖÕ.
(230) u.
(231) 60I0 D. [ai ]q ∩ [ai + xi − 1]q. 0/= D . , '1 '. F 6=0% :. D [a ] ∩ [a + x − 1] ⊆ L(v ) i i q i i q j 6 09- 0'. .
(232) 04
(233) % 6 & 60&/"0 60I0 7 vj : xi − 1]q. , "&0876. L(vj ) \. S. i [ai ]q. ∩ [ai +. ¶§¶ oJc r^P
(234) _aP jaOf{eNPUKNfgbK°bc#z|c#dNj]Uc hjoPP
(235) ¦!_oP
(236) erhNz{P0;f_]`6d Eb
(237) P ]n_ac b
(238) c0e]afgr^P
(239) j>h0z|zqc!]a]af{Nz|P
(240) b c0z{c0dNjof{eN#]3c0§_aKP~¢#P
(241) ja_afgbP ]kc F Uf|_aKbc#z|c#vdNjo]bKNc#]aP
(242) e5¨jac#O _oKNP
(243) f{jjaP%]`q;P b_af{¢0P>P
(244) ¦,_aP
(245) erNhNz{P>z{fg]m_] E ; f > C4@\¯_aKNPbc#er^f|_af{c0ec0e_oKNPBc#jor^P jaf{eNxcL_oKNPB¢#P
(246) ja_afgbP ] "h)bc#z|c#dNj c fg]eNc0_P¦,_aP erNh0Nz|P ¨c#j v f|nher¥c#eNz|\°fUf|_f{] f{e¥_aKNPf{e!_aP jo]aP b_af{c0e T [a] ¨c#j]ac0OP i JLKNvP ]aPf|e!_aP jo]aP b_af{c0e]`P
(247) _o] h0jaP Oh¦^f|Ofg]`P%r¬UKNP e _oKNP E hjoPBf{e!_aP
(248) jo¢hzg]
(249) k0f{¢!f{eN | T [a] | = 2q − x kf| x < 2q kher T ^c_oKNP
(250) joUf{]aP0 | ; > [a] | = 0 f|f C3Z,f{ebP x < 2q f|_¨c#z|z{cn]¨joc0OI_oKNP jo]`_3bc0er^f_of|c#e _aKh_ [a ] ∩[a +x −1] ⊆ T [a] JLK,d] ^r^dNP~_acS_aKNPc#]`P ja¢h_af{c0e5Oh0rNP>f{eB_oKNP~qjac,cc ; f > _ac#0P
(251) _aKNP jUf|_aK _aKNPWzgh0]`__mcSb
(252) c0er^f|_af{c0e]
(253) _oKNP qNjoc,cKc0zgrN]
(254) . j. i. j. j. q. a∈Ei. i. a∈Ei. q. i. . i. i. q. a∈Ei. i q. i. i. q. a∈Ei. q.
(255)
(256) . ¡c0jWhe,\5_mc5q;c#]af_of|¢#Pf|e!_oP
(257) 0P jo] a ≥ 2b _oKNP#joh0qNK K .b
(258) hz{z{P r_aKP
(259) : +!0 .Kh#]n¢0P j`_oP¦ ]aP_ {0, . . . , a − 1} her ij f{]~hexP rN0P f|3h0erxc0ez|\ f b ≤ |i − j| ≤ a − b pnP
(260) Ohjo®_aKh_ ¨c0j~he,\ e |u% .;K,dªqNjoc¢0P%r5_oKh_ ¨c#j~h0e!\ k ≥ 1 her k≥1 K ) = 2+ h#]`®#P rxUKNP
(261) _aKNP jW_aKNP'bCf{job
(262) dNzghjWz|fg]m_ bKNjoc0Oh_of{be!dO;P
(263) jc cch(C g f ] L J N K g f ~ ] g f ~ ] N e c_W_aKNPb
(264) h#]`P#]`f{ebP _oKNPbf{jbdNzghjUbz{fg£#dP ]Ubc#e!_ohf{ez{h0ja#PWb
(265) c0OqNz{P_aP f|qh0j`_of_oP ]`KdN0jhqNK] "Vªc0joPqNjaP%bfg]`P z|\# (¶ %k¶ Ä ¶ #" '
(266) 33& 0 ' 0D()0 00 09-.,& 36%D 0 '1 0 ()0% N a, b 8 a ≥ 2b a/b. 2k+1. (2k+1)/k. 1 k. 2k+1 a a/b b. &% 6 60
(267) . $ 00' %040 00' 8. cch(Ka/b ). '.. a b. ,I/4 0 6 '. N. :. * ]fg]GP
(268) z{z#®,eNcUeG ¶§¶ o7 #GP_ X ]`P P>¼u #GP_ 0h er b = k #¨mc0j≥]`c#OQPUq;c#]af|_af{¢0Pnf{e!_aP
(269) #P
(270) j k > 2m "JLch(K KNPn#joh0qNK )K≥ N +4 bc#e!_ohf{e] K h0] ah=]`dN2k+2m 0jhqNKGC _h®#P)_aKP¥¢#P
(271) ja_afgbP%] {0, 1, . . . , m − 1} ∪ {k + m − 1, k + m, . . . , k + 2m} &%nP
(272) ebP0WP¯Kh¢#P '%ncP
(273) ¢#P
(274) j%f k > 2m _aKP
(275) e = F <3 cch(K ≥ ch(K )−1 ≥ ch(K )−1 ≥ N +3 _oKNP
(276) joP¨c#jaP#) cch(K )− >N . . 2(N +3)−1 N +3. m,m. a/b. a/b. a/b. a/b. (. a b. m,m. a b. m,m. 2k+2m k. )+*, .-/0 . ]3OP
(277) e!_af{c0eP rh;c¢0P##;K,d{%uqNjoc¢0P r_aKh_3_aKPnbf{job
(278) dNz{h0j bKNc0fgbPLe,dNOSP j c_aKNPnc^rNrb
(279) \,b
(280) z|P C JLKNPWÄh0b__oKh_ cch(C ) ≥ 2 + f{]_ojaf{¢,f{h0z]`f{ebP~_aKPb
(281) f|jbdz{h0jbKNjoc0Oh_afgb>e,dNOSP jc0 2+ g f ] C ¤¥P]`K2h+z{zd]`P_oKNP¨c0z{z{cUf|eNQz{P
(282) OOhz{h_aP
(283) jUc#e§ 1 f{¢0P e¥h 0jhqNK Gh '/.2
(284) 0 c0 fg]he¥f|erNdbP%r°qh_aK§Uf|_aK¥q;c#]o]`f{Nz{\x_aKNPQ_mc P er^¢0P j`_of{b
(285) P ] ;P
(286) f{eNQ_aKP]ah0OQP¢#P
(287) jaG_aP¦^UKNc!]`Pf{e!_aP
(288) joGf{c0jL¢0P j`_of{b
(289) P ]Lhz{z§Kh¢#PWr^P 0joP
(290) PW_mcSf{e G *. fg]. 1 k. 2k+1. ÌÌ ÏKL«÷&LM. 1 k. 2k+1. 1 k. 2k+1.
(291) 04 '2(.
(292) 0
(293) 00'1 J. ² ¬ 0. : 4 3 330. - 33&D 30 ' 0D()0. G {v1 , . . . , vn }. : ; 0 %0 , n '/.2
(294) 0 L <6 v% 0 'v1 v%0 2 ·09·-· v 0n'v.n+1 09. 0=0'D0<( H 36 :. . 0$' (p, '1q) $. 2 n. 2: + G. . +,& &(.'/"0'1<76= 60(. 36 $ 3 0%%
(295) &')( 76 (p, q) $ L $ G\. ¶§¶o #§P
(296) _ Z,f{eb
(297) P Kh0]c#eNPxP¦,_aP erNh0Nz|Pªb
(298) c0z{c0dNj% Kh0] ,\ #GP
(299) OOh ; f > Lh_ z{P h#]m_ tq − 2qt+=1 2=+ + 1 P¦,_aP evrNhz|P5b
(300) c0z{c0dNj]
(301) erNdb«_of|¢#P
(302) z{\0GP5vb
(303) h0e°_oK,d]]aKNc _oKh_ v Kh0] h_z|P%h0]`_ i + 1 P
(304) ¦,_aP
(305) erNhNz{P5bc#z|c#dNj]
(306) ¨c#j i < n c v Kh#] c0eNPP
(307) ¦,_aP
(308) erNhNz{P5bc#z|c#dNjher Kh#]h_z{P h#]m_ (n − 1) + 1 P
(309) ¦!_oP
(310) erhNz{P5bc#z|c#dNjo]F_aK,d] ,\ #GP
(311) OOh ; f > v Kh#]h_z{P h#]m_ v P
(312) ¦!_oP
(313) erhNz{Pbc#z|c#dNj]
(314) N]ac_aKh_n_aKPjaP%£!dNf|joP rxb
(315) c0z{c0dNjof|e tq − (2q − 1) − 2q − (n − 1) − 1 = 2 f{er^P P rP¦^fg]m_]
(316) * ]nhr^f|joP b_>bc#jac#z|zghjo\cG_oKNPh;c¢0PWz{P
(317) OOhNNP0P7_ C ¶ ¶ Ä^ ³ 00& '1 0 ()0 n ≥ 3 I0 . 50 C 6
(318) %69%0 ' /"%0 I/"3& 2 n 2q n. 2q n. 0. i. n+1. 2q n. n−1. 2+. 2 n−1. 1. n. 2q n. :. n. JLKNPQdNqNq;P
(319) jW;c0dNer°0f{¢0P e),\)Rc#jac#z|zghjo\ H f{]Wc0qN_af{Oh0zG¨c#j~c^rNr°b\^bz{P ] @d^_W_aKPS]af_odh_of|c#eª¨c0j P ¢0P e b\^b
(320) z|P%]]aP
(321) P O]Oc#jaPbc#OQqz|P
(322) ¦. %>P
(323) joP~fg]UhSb
(324) c0e]`_ajodb_af{¢0PWqNjac,c0Gc§_oKNP bf{job
(325) dNzghjLbKNc0fgbPWe,dNO;P
(326) j ¨c#jL_aKNP]aOhz{z|P%]m_UP ¢0P
(327) exb\^b
(328) z|P# : (¶ %k¶ Ä ¶ cch(C ) = 2 ¶§¶ oZ,dNjoP
(329) z{\ #§P
(330) _ her r^P eNc_oP_aKNP¢#P
(331) ja_afgbP%]c §_oKNP5¢0P j`_oP¦ ;P
(332) f{eNBh#rl`h0b
(333) P
(334) e!_Ucch(C _ac v h)er≥v2 U¡f|¦_mv,cvf{e!,_avP
(335) #P
(336) j] pvh0er q h0erªh 2© (p, q)© z{f{]`_ © h0C]o]`f{0eOQP e!_ L L¤¥vP ]aKh0z|zqNjoc¢0P~_aKh_ C f{] (p, q)© L© b
(337) c0z{c0dNjhNz{P0 ¡f{jo]`_ ;jaP Ohjo®_aKh_~f| ) ∩ L(v ) 6= ∅ c#j L(v ) ∩ L(v ) 6= ∅ _oKNP
(338) e)_aKNPrNP ]af|joP rªb
(339) c0z{c0dNjof|e P
(340) ¦^f{]`_o] Cnlmd]`_Qh#]a]af{0e¥_aKNP ]ah0L(v OQP bc0z{c0dj _ac"]oh\0 v h0er v c v her v P h0bKKh¢#P5h_z{P h#]m_ b # c | z # c N d U j h ¢hf{z{h0Nz|PWUf|_aKªr^fg]m_heb
(341) P _ocQ_aKNPb
(342) c0z{c0dNjnbKNc!]`P e5¨c#j v her v 2q − (2q − 1) = 1 Z,cP ]`dqNqc!]`PeNc _aKh_nc0_aKxf{e#_oP
(343) j]`P%b«_of|c#e]Lhjo>PWqP Oq^_m\0 #GP_nd]UqjaP%bc0z{c0dj Uf_oK)hQb
(344) c0z{c0dNj #§P_ ;P _oKNP e,dNO;P
(345) j>b
(346) c0z{c0dNj]c ) _oKh_ h0jaPeNc0_P¦,_oP
(347) erNh0Nz{P0Gvf@ P0§_oKNP5e,dNOSP jcaU∈bc#L(v z|c#dNjo]) f{e L(vx¯ ) ∩ [a] * ] L(v ) ∩ L(v L(v _aKNP ) =∅ e,dNOSP jUc3bc#z|c#dNjo]c0 L(v ) _oKh_>h0jaPeNc0_LP
(348) ¦,_aP
(349) erNhNz{P fg]Uh_UOc#]`_ 2q − 1 − x¯ . c PLb hehqNqz|\ _oKNPLz|P OOhWc#ebPUh!hf{e§. ~P
(350) ec_aPL,\ _aKNPLe,dNOSP j3c.P¦,_aP erNhz|PUbc#z|c#dNj] c0 v f{e L(v ) ¨c0j i ∈ {2, 3, 4} JLKNP ¢#P
(351) ja_aP¦ v fg]nh0rl`h#bP e#_Lx_oc v h0er v her]af|eb
(352) P x = 2q − x¯ h0er x ≥ 2q − (2q − x¯ − 1) = x¯ + 1 PUf{e^¨P
(353) j _aKh_ x ≥ 2q − (2q − (2q − x¯)) − (2q − (¯x + 1)) = 1 4. 4. 1. 2. 2. 3. 4. 4. 1. 3. 4. 1. 4. 2. 1. 3. 4. 2. 3. 1. 1. 4. 1. 2. 2. q. 2. 3. 3. i. i. i. 3. 4. 2. 3. 2. 4. . ¶
(354) ² G ² . 604
(355) %6 90:'1 /4%0 <60"30& 0"30' .
(356) 0=, 8. :. ë¨î̧ëÖÕ.
(357) w.
(358) . ! L " G x ; # ! L L. U pnP b
(359) h0z|zG_aKh_>_oKNP .0D()0' 0 . ckh#joh0qNK f{]n_aKNPSOh¦^f|OSdNO c¢0P
(360) jnh0z|z"]adNN0jhqK]Uc c _oKNPOf|eNf{OdO r^P
(361) #jaP P0"¤¥P]ohδ\Q(G) _aKh_ G fg] k $ ..0 (0G' 0 0 f| δ (G) ≤ k 3JLKNPW¨c0z{z{cUf|eNQjoP ]adNz_LGLh0] ]aKNcUef|e¥{%u% C
(362) = " cch(G) ≤ 2δ (G) : 0² ¬ JLKf{]>jaP%]`dNz|_~Lh0]>hzg]`cB]aKNcUeª,\);K,d |u%_ocB;PSh#]`\,Oq^_ac0_afgb
(363) h0z|z{\5_of|#K#_W,\ ; P ]o]`P e#_of{h0z|z{\ > ]aKNcUf|e _oKh_ _oKNP5bc#OqNz|P
(364) _aPNf{qh0j`_of_oP0jhqNK K UKf{bKKh#] r^P
(365) #P
(366) eNP joh#b\ k ]ah_afg] P%] cch(K ) ≥ JLKNfg]Uz{P rª;K,d _ocqc!]`PW_aKP¨c0z{z|cUf{eNqNjoc0Nz{P
(367) O C k 2− ² > 00 '6& ' α &% 6 6:00&:( 36 G cch(G) ≤ α ch(G) ¶ . c_aP_oKh_nf| P¦^fg]m_]Lf_nfg]nh_Uz{P h#]m_L_mcNh0] e_oKNf{]]`P%b«_of|c#αe§0P j]m_kh0e]`P
(368) j3f{e_aKP>h EjoOch(K h_af{¢0P_ac )ik≤jac#Nδz|P (KO W¨c#j ) 2+© b1KNc,=c#]okh+z|P10jhqNK]
(369) e _oKNfg]Ub
(370) h0]aP0!P]aKNc _oKh_L_aKPb
(371) c0jojaP%]`q;c0er^f|ebc#e]m_he!_ α f{]Lh_LOc#]`_ 5/2 "JLKNP e§^Oc0joP>0P eNP
(372) jhz{z{\0 P ]`Kc _aKh_ cch(G) = O(ch(G) + ln(|V (G)|)) N¨c#jnhe,\50jhqK G . ∗. ∗. ∗. k,mk. k,mk. 2k m. ∗. k,mk. . 2. k,mk. , "!, #$ . JLKPkhf{OIc,_aKNfg]]`dN]`P%b«_af{c0e fg]_oc~]aKNc¯_aKh_P
(373) ¢#P
(374) jo\ 2© bKNc,c!]ah0Nz|P"#joh0qNK Kh0]bf{job
(375) dNzghjGbKNc#f{b
(376) P3e,dNO;P
(377) j h_ Oc#]`_ 5/2 JLKNP jaP
(378) ¨c0joP0#PUd]aPU_oKNP>bKh0joh#b«_aP jafg]oh_af{c0e_aKNP 2© bKc!c!]ah0Nz{P#joh0qNK]3,\ kjor!]
(379) !pUdNNf{e h0erJh\,z{c0j u@ JLKNP 0 & chB0jhqNK fg]>_oKNPOh¦^f|Oh0z"]adNN#joh0qNK)f{e°UKNfgbK)_oKNP
(380) joPQf{]eNc¢0P j`_oP¦ªcrNP
(381) 0joP
(382) P c#eNP0 * zg]ac;_aKNPQ0jhqK°bc#e]`Gfg]`_af{eN5ck_mc ¢0P
(383) ja_afgbP%]>b
(384) c0eNeNP%b«_oP r)Uf|_aK°_aKNjoP
(385) PSf|er^P
(386) q;P
(387) erNP
(388) e!_ qh_oK]>c0 z{P
(389) e_aK i j h0er k fg]nr^P
(390) ec_aP%rB,\ θ , /.021 %'1' 0% 0 . ( 36, 2 $ 508 6- '. ' % ²,¶ ² &')(>+· * ) · % ³ ¶ i,j,k. 6 <60% & ,0 60= "%'2( 50 0& 09- ' 0"30' .
(391) 0= . θ2m,2,2. 6 . /"0I'1 0 ()0. m≥1. :. ¶(%k¶ Ä ¶ 13 '1 m ≥ 1 cch(θ ) ≤ 5/2 : ¶§¶ o #§P
(392) _ her xwy ;P_oKNP_aKjaP Pqh_oK]¨c0joOf|e θ #§P
(393) _ L PBh xu . . . u y xvy | z g f ` ] _ 0 h o ] ` ] { f 0 N e O P # e _ c k¤¥P qNjac¢#PW_aKh_ θ h#r^Of_] (p, q)© L© bc#z|c#dNjaf{eN!UKf{bK 5/2© (p, q)© fg]U]`_ajoc0eN#P
(394) j_aK© h0e_aKNPr^P%]`f{joP rθbc0ebz{d]`f{c0eG * ]a]adNO P jo]`__aKh_ ._aKP
(395) e h0]o]`f{0e)_ac u v her w _aKNP 6= ∅ ]ohOPb
(396) c0z{c0dNj α P
(397) ¦!_%mh0]o]`=f{0e1 _ac xL(uh0er )y∩h5L(v) bc#z|c#∩dNj>L(w) f{e L(x) h0er)h bc0z{c0djnf|e \ [α] joP ]aq;P b«_of|¢#P
(398) z{\0JLKNfg]>\,f{P
(399) zgrN]~_aKNPr^P ]af{jaP%r)bc#z|c#dNjaf{eN keNc0_ §h#]a]a\f|#[α]ex_ac x h bc#z|c#dNj α f{e L(y) I#GP_ L(x) h e r , Z { f e b P iPk= ) ∩ [α] | i = |L(v) ∩K,d[α] | i r = K |L(w) ∩ [α] | L(u ) ∩ L(v)he ∩r L(w) 6= ∅ Kh|L(u ¢0P i +i G L J ] h e h 0 ¢ P h _ | z % P 0 h ` ] _ 5q/2−i P
(400) ¦,_aP
(401) erNhNz{PSb
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(403) e,b
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(406) ¦,_aP
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(408) c0z{c0dNjhNz{P0 . 2m,2,2. 1. 2m−1. 2m,2,2. 2m,2,2. 2m,2,2. 1. 1. q. u. 1. u. q. v. w. v. q. w. 1. q. q. 1. u. v. u. 2,2,2. ÌÌ ÏKL«÷&LM. w. v. w.
(409) . 04 '2(.
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(412) c0z{c0dNj α P¦,_ h0]o]`f{0e°_ac x h mb
(413) ≥c0z{c02dNjWc L(v) 4@\ #§P
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(415) ja_aP
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(417) ¦,_aP
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(419) erN\h0[α]Nz{Pb
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(421) ¢#P
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(427) z{≤\0 2q* −]Wh1;c¢0P0 Kh0]h_z{P h0]`_ q/2 P
(428) ¦,_aP
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(430) ¢#P
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(432) erNh0Nz{P5bc0z{c0djo] BRc0e]aP £!dNP e#_oz|\#h0#hf{e°,\ #§P
(433) OOh0 y Kh#]h_ z{P h0]`_ u P¦,_aP erNhz|Pbc0z{c0djo] %>P
(434) eb
(435) P0 θ f{] 5q/2 − (2(2q − 5q/2) + i + i ) − (2q − 3q/2) ≥ 1
(436) b 0 c { z 0 c N d j h N { z 0 P (p, q)© L© ¶ ¶ Ä^ ³ 11 30& 2 $ 50 (. 36 6 3
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(438) P # ¢
(439) P o j \ { z g f m ] _ # h a ] a ] | f # N e O
(440) P ! e _ c b N K , c # c o ] h N { z S P
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(443) joP
(444) ebPn2; © P_mP
(445) P e_aKNP~e,dNO;P
(446) jc¢0P j`_of{b
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(448) jUc0¢#P
(449) ja_afgbP%]c0f|_o]UKP hja_ x Rz{P h0jaz{\0§f| PKh¢0PS_aKPjaP%]`dNz|_!\Rc0joc0z{z{h0ja\ Hªher¯i joc0q;c#]af_of|c#e %NZ,dNqNq;c#]aPeNc _oKh_ x ≥ 1 "JLKNxP
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(457) _aKNP j cch(G) = O(ch(G)) "¤¥PLh0jaPeNc_ h0Nz|P_ac]aP_`_oz|P_oKNPU£#dP ]`_af{c0eKP
(458) joP0Nd^_ JLKP
(459) c0joP
(460) O %}Q;P
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(462) Oq^_ _occ0q^_of|Ofg]`Pb
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(464) 30& 9%0 A B.I % ²,¶ ² 1 " '1( 36 G 8 n cch(G) ≤ 36 ch(G) + 54 ln n + 3.. ¶§¶o;¡f|¦ h0er)]aP_ WZ,dqNqc!]`P_aKh_~z{f{]`_o] ⊆ Z ck]af PSh_ z{P h#]m_ tq + 1 p,hjoqP#f|¢#P
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(466) e°+PS54b lnhe°nb
(467) +P
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(472) P2}e!dO;P
(473) j]
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(477) z{P
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(482) j]
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(487) joUfg]`P##P~bKNc,c!]`Pnc#eNP>c0 3i + 1, 3i + 2 h_jher^c0O P h#bK5Uf_oKBqNjoc0hNf{z|f|_m\ A #§P_Ld]r^P
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(489) v.
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(491) _cG]aP
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(493) ¦ k ∈ n0, . . . , j ko P~h#]a]ac^bfgh_oP>he f|e!_aP ja¢h0z I = {k(q − 1), . . . , (k + 1)(q − 1) − 1} c Z c_of{b
(494) P>_oKh__aKNP I hjoP rNf{]Älmc0f{e!_Lf{e!_aP ja¢hzg]Lcz|P eN_oK q − 1 * bjodbfghz.c0]`P ja¢h_of|c#e5¨c#jL_aKNP]aP £!dNP zf{]_oKh_%Nf k, l ∈ K h0jaP rNf{]`_af{eb«_>h0er a ∈ I , b ∈ I N_aKNP e |a − b| ≥ q kZ^P
(495) P¡f|#dNjoPS# p−1 q−1. k. k. p. l. k. p. I. p−1. 0. ... 3(q − 1). 6(q − 1). 9(q − 1). j. ···. p−1 3(q−1). k. (q − 1). ¡ f{0djaP.C * e f{z{z|d]`_ajh_of|c#e c0G_oKNPÔ_oKNf{eNeNf{eN qNjoc,b
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(497) OY}^ S #GP_nd]n]aP_ ¡Nc#jUP h#bK v ∈ V Nz{P_nd]Ur^P eP I I= k∈K k. S(v) = {k ∈ K : Ik ∩ L(v) 6= ∅}.. JLKPnf{rNP h ¨c#j _oKNP>joP ]`_kc._oKNP>qNjoc,cfg] _oc]`Kc _oKh_ t h#]kbKNc!]`P eQf{eB]adbK5h Lh\_oKh_ P(|S(v)| < ¨c0jnh0z|z v _nUf|z{z_oKNP
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(499) jc0jrN] _aKNP jaP Od]`_SP
(500) ¦,fg]`_Qh)bKNc#f{b
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(502) P#G¨c#jUKNfgbK ¨c0jSh0z|z v ∈ V @\°_aKNP rNP eNf|_af{c0e c0 ch(G) "_oKNP
(503) joP5OSd]m_ P
(504) ¦^f{]`_Qh¥bc#z|c#dNjaf{eN c |S(v)| c G U≥f|_aK ch(G) #§P_d]r^P eNPxh)eNP bc#z|c#dNjaf{eN f ,\ bKNc,c#]af{eN c(v) ∈ S(v) f . L J N K g f ]
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(506) c0z{c0dNjof{eN;P b hd]aPf vw ∈ E(G) _oKNP
(507) e c(v),v c(w) hjoPr^f{]`_af{eb_P
(508) z{P
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(513) e¨joc0O eNc#e © h0rl`h#bP
(514) e!_kf{e!_aP ja¢hzg] I , I ^]aKc _oKh_ |f (v) − f (w)| ≥ q _LjaP Ohf{e]3_ocS]aKNc _aKh_ t fg]bKNc!]`P e5]`dbK_aKh_ P(S(v) < ch(G)) < KNc0zgrN] ¤¥ P j]`_bzghf{O _oKh_U_oKNP qNjoc0h0Nf{z|f|_m\_aKh_ |S(v)| f{]nc#dNer^P%rh0c¢#P~,\ < ch(G) 1 n. k. c(v). p. c(w). 1 n. . . 1 P Bi s, 6. . . ≤ ch(G) ,. UKP
(515) joP 1 exc0jr^P
(516) jU_ocqNjoc¢0PW_aKNPSbzghf{O j]m_neNc0_aP_oKh_>Pb
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(526) z{P
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(545) c¢0P jaP%r0f{¢0P e_oKh_ A(i, J) KNc#z{rN]"fg] h_"z{P h#]m_"_oKN∪PUIOf|eNf{OdO c0"_mcqNjoc0hNf{z|f|_af{P ] C _oKNPqNjoc0hNf{z|f|_m\5_oKh_ 3k f{]~bKNc#]aP
(546) ex#f|¢#P
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(548) eª#f|¢#P
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(551) PW¨c0jUf{e]`_oh0ebPÅw% ]m_h_oP ]_aKh_ 0. 0. tq 3(q−1). 0. 1. l. i. i+1. j. j. i. 1. i−1. l. i. 3k+1. i. 3k+2. 1 3. i. 1 3. k−1. i. 3k. i−1. 3(k−1)+1. 3(k−1)+2. i. k−1. k−1. 1 6. JLKf{]Uh0z|z{cn]d]L_acUjof_oP. P(Bi(k, p) ≤ kp − r) ≤ e−2. r2 k. .. s 2 /s . P(S(v) < ch(G)) ≤ exp −2 ch(G) − 6. Z^c^PWc0dz{r z|f{®0P_ocKh¢0P c#jLf|exc_oKNP
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