Fractional Coloring of Bounded Degree Trees
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Fractional coloring of Bounded degree trees Afonso Ferreira — Stéphane Pérennes — Hervé Rivano. N° 4094 January 2001. ISSN 0249-6399. ISRN INRIA/RR--4094--FR+ENG. THÈME 1. apport de recherche.
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(447) R1o`]recRp]rs3a[V: k ? Q
(448) B7 6 E = qB e
(449) B756 b
(450). r_rtsuqvwxv.
(451) T+w.
(452) ¿¿ ¿[À.
(453) ¿
(454) ÁPÁ. e ; [ . 6ÀnÁAÅ. U. ¬ °Jf^e Z[f^QSRj
(455) f^]rmPRif^\^KR1Z&?
(456) QB>6 E ¥JsPj
(457) ]Porsr~»a[MrR E\^KR1© : k ?5
(458) eB76 E PNa[MrR1s ZksPjR : k ?5e
(459) B7 6 E = : P ? Q
(460) B> 6 E a[MrR1e[RSR_`Z'aqBZ k ' k ?5Q
(461) B> 6 ED\nsP o P ' P ?5
(462) eB76 El a[M > k ?
(463) k E ¢\nsKo > P ?
(464) P ED r p R
(465) a > <=$ ?> k ?
(466) k E B3> P ? P E Ec p R
(467) Va = k n k ?5Q
(468) EB k ?9 6 E3 P 3 P ?5Q
(469) EB P ? 6 E3 >e
(470) B7#6 0)\nsKoisrfnacR°a[MK\ aB ZD\^ssKo`R mKR1sPo`R1s0aDZ[Ralf^ 0 ¬sPjecR1\0ZkR >?
(471) YE:{b > orR1jecR1\0ZkR > k ?
(472) k E:\nsPo > P ?
(473) P E{Cb > ª Lf^R1e[JbSa[MrRj
(474) \^Q¢ g']KZ'alsrf^a[RVa[MK\ a a[MPR°mre[f`j
(475) R1o`]recRpmrecR1Z[R
(476) ec^R1Z:a[MPRpJf^f sr~s{ \nec\^s3a b Sa[eq\^j
(477) RV R 3]P\na'bC¥ : k = : P © Id\ a[MKZl\necRVj
(478) f ^R1e[R+oIR
(479) a[MPR
(480) e {b > k > P f0_e > G
(481) ?>6Eδ G
(482)
(483) ?
(484) > k E =3G
(485) ?> P E)´ G
(486)
(487) ?
(488) >6Ea=3G ¬ a Jf0f lZa[MP\na1P\na a[MrR°R1sPoª8 > Zl\Jec\0jacf0sP\nªjf0f0e[sr~fnM ?F+0 B +E@ a[Mijf3Z'a:G `Qf0e[R1f ^R1e10Jf^e a= ³ B a[MrRpacec\0jRVfn+ > f^s¢\ns{bIR1or~^Rpfn 0 R 3]P\nZa[MrR°f0srRpfn+ >sX0 ' Ì Ì # #4` É NÁ\SÂqÀ ÁÂqÅR: 8AÆ À 8rÅ ] Á)Å:O4À =<3O Á cÀn¿ 8A¿[Å n¿[ÀR: \Ã{Å;4
(489) Å Æ 4Å ÁYÀ ¿ 4U6ÀR 4Ä ®¿[À0Â Åo F;9;:4
(490) ÁYK°Å OE
(491) o? ?F0 B EB7GE ÌÎ! Ì 6 RD KecZka\0Z[Z[]rQSRlacMP\ a Zk]KjqM\mPe[f0~^eq\nQ±o`f{R1ZR_`ZkaJf^eKf0]rsPo`R+o o`R1~^ecR
(492) RDZkac\^ecZ13\sP\^0R N\6bVa[fp~^R
(493) a@f^srRZ acf¤]KZk/R À3¾ ;9qFÄ ÁAÄo8qÁ)Ä
(494) ÁY_;eU À ¿4À3¾V0a[MP\naZf^sPR \nec\nrR:Jf0e@\ns{b°mKf3Z[Z[rR \nAR
(495) R1oCsPo`R
(496) mAR
(497) sKo`R
(498) s3aZkR
(499) a°s\ Af^]PsPo`R1oEo`R1~^ecR
(500) RSZ'aq\ne¥4\0ZVormP\na[MPZ°\^e[R\nAR
(501) R1oªÎRQ]PZkao`Zka[s` ~0]rZkM KR
(502) a'R1R
(503) s¢ZkQS\ne o`mP\na[M¢MP\6{sP~o`)R
(504) ecR
(505) s3al\^KR1Zq© `DR
(506) sPj
(507) R^racMrRpf3\^oMZ j@K\ a QSf0Zk/a ?KjaEP or)R
(508) ecR
(509) s3a\nAR
(510) R1o sKo`R
(511) mAR
(512) sPorR
(513) s3aNZ[RaqZj1\ns j
(514) f^ece[R+ZkmAf^sKoa[f\^sI]rsr\nAR
(515) R1o sKo`R
(516) mAR
(517) sPorR
(518) s3aNZ[Ra+ `lR
(519) sKjR¤\ mAR1ZcZ[QSZ'acjR1Zka[Q\ a[Rjf0]rs3ap\^Kf0]`a ?qEjQP \nec\nrR1Z1) MrR
(520) ecR ?qENZ¤acMrRs{]rQKR1e¤f^:mAR
(521) e[JR1ja Q\nacjqMrsr~0Z s i½req\^ja[f^sP\^jf^f^ecsr~¢Za[ec{\nbEo`R1ZcjecAR1oEJecf^Q"a[MPZ6\^e[\nPR+Z
(522) <a[MrR1s a[MrR acec\0jRp \nec\^rR1ZN\necRVZ[QmPRZ[]rQ f^ Z[]rPZ[RaqZNfn a[MrR°\nAR
(523) R1o sPo`R
(524) mAR
(525) sKo`R
(526) s3aDZ[Ra \nec\^rR+Z
(527) . RIacMrR
(528) s ]PZ[R \^s sPor]Pjac0R \n~^f0e[a[MrQºa[f~0R
(529) srR1ec\na[RI\isrR1\^emrecf^~^eq\nQº Mrf0Z[R Z[f^]`a[f^sPZ\necR acec\0jR+ZpfnOE
(530) ??F0+B +E B7GnE[ ¤Z[Z[]rQSRa[MK\ a0j
(531) \^sKR Z[mras3a[f ?F0 k B k EV\nsPo ?0 P B P E¤\^sPoER
(532) aE ARacMrRsrR1\^eDmrecf^~0ec\^Q Jf^eDa[eq\^j
(533) R1Zlf^]E
(534) ??F0oB EB>GnE[A MrR
(535) ecRR\0Z[Z[]rQSR°a[MK\ aDacMrRacec\0jRf0s¹Z \0Z[Z[f`j\ acR1oacfa[MrR° \nec\nrR1ZN: ?5e
(536) B7 6 E[LNMPR
(537) s¢\Zkb`Zka[R
(538) Q Jf^e6?F0 B EZ6b. r.
(539) ο[À0Â4Å Á)ÀÂqÅÅ ¿ Á ¢Å
(540) DÅnÆ`ÁAÄcÄ Ä^¿®¿. E = E k. . T0T. <
(541) eB76 E P : k ; < D = : P ; . Ì 7 Ì , ,JÌ* 2 ª§? ¿qÄ ÆPÂXÅ4
(542) Å#UnÀ68rÅ ] ÁAÅ:O4Àe=<J Á qÀ ¿M8K¿cÅ0¾9: X ÌÎÌ!6 RDZ[QSmrbMK\6^RNa[fZkMrf Mrf a[fjf^QSmr]ra[RD\^sR
(543) R
(544) QSR
(545) s3a:f^ E
(546) ? ?0+B EB>GnEªJecf^Q±\pa[eq\^j
(547) R f^A\nsR
(548) R
(549) QSR1s0af^ E
(550) o? ?0+B EB>GnE¥Jf^rac\nsrR+oJecf^Q jf^ecf^\necbp|0© l~3\nsRNmre[f`j
(551) R
(552) R1osKo`]Pjac0R
(553) b%b RZkac\^ekaJe[f0Q Z'aq\neqZ\^sPop PsPoJf^e R1\0jqMf^sPR\ Jeq\^ja[f^sP\^3j
(554) f^f^ecsP~lMP\6{sP~ a[MrRa[eq\^j
(555) RacMP\ a sPo`]Pj
(556) R1Z f0sIa1LNMrR1s ?F0 B E j
(557) \^s ARVZ[mra s3a[%f ?0 k B k E\^sP& o ?F0 P B P E[`mPe[f0mKf3Zka[f^sT¤mrecf 3o`R+Z\\6ba[f QSR1e[~0RJeq\^jacf0sP\njf0f0e[sr~3ZDf^ ?F0 k B k EN\^sPH o ?F0 P B P E MrR1sacMrR
(558) eVa[eq\^j
(559) R1ZV\necR R 0]K\n®)\^sPoiacMrZpZ acMrRj
(560) \0ZkRpZ[sPj
(561) Rpa[MrR°~0f0P\nAacec\0jRpZN \^oª.
(562) '7)+/#% MO ,- ' ¼DfnacRa[MK\ a¤f0]rel KecZka¤QSf`o`R
(563) sPo`]Pj
(564) R1Z¤^R
(565) ecb \^e[~0RZkb`Zka[R
(566) QZ¤ZksPjRRjf0]ro¢~0Ra6NO?k³RE$ j \^e[\nrR1Z Jf0eDrsP\^e[ba[ecR
(567) R+Z
(568) A\nsPo NO?[R³ E$ jJf0ela[R1e[sK\necb f^sPR1Z1l¬sia[MPZVZkR+jacf0siRZ[Mrf Mrf a[f e[R+o`]PjRa[MrR Z[b`Z'acR
(569) Q²Zk£
(570) RSacf ?qEjQP;$@a[MP\naZ\iZk£
(571) Rf^ f^eqo`R
(572) 6e $ jQP3NO?kR³ EDJf^eAf^]rsKo`R1o o`R1~^ecR
(573) Ra[ecR
(574) R+ZI¥J a[M Zka[<\4\^ZkaN~0e[f sr~Sjf0sPZ'aq\ns3aq© Ì 7 Ì , ,J Ì * C ª§? ¿ qÄ ÆP X4Å
(575) Å #UnF \Á qÀnS¿ 8K¿[Å3¾ V: \ Ï%Ã =<3 ?qEjQP
(576) $X ÌÎ! Ì 6 ½ ecZka\nsPoC acMrf^]raf3Z[Zpf^~^R1srR
(577) eq\na'biR\^ZcZ[]rQSRacMP\ a°acMrRf3\^oZp]rsPJf0e[Q ¥Jsrf^a1 R°\^oroR
(578) sr~^a[M³°o`mK\ a[MKZ MrjqM o`fSsrf^alQSf{orJb a[MrRpJeq\^ja[f^sP\^)jqMPe[f0QS\na[jVs3]PQAR
(579) e© ¼Dfna[R:a[MP\na<acMrRZ[b`Z'acR
(580) Q a[MP\na o`R+Z[j
(581) e[AR1ZÎ \^opa[eq\^j
(582) R1ZÎj1\nsAR\^acR
(583) ecsP\ ac0R
(584) b¤j
(585) f^sPZ[orR
(586) ecR1o°\^ZÎJf^f b \nec\^rR1ZB:; Y< m Jf0e¤J¹ ?5Q
(587) Ef0e!)
(588) = I\nsPoiJ¹ ? 6 Ef0eB6 = o`R+Z[j
(589) e[ARpa[MrRa[eq\^jR°acZ[R
(590) ' ¥Jsrf^a[RpacMP\ a Jf0e!
(591) =D S\nsKoJ 6 =3 q BN: ; ^< m o`fSsrf^aDo`R
(592) mAR
(593) sKof0s acMrR°R1or~^R°jf0sPZko`R1e[R+oP© I½rf0e R+\^jqM Zkac\^ef^a[MPRace[R1RacMrREorR1Zcjecm`acf0s f^\»f`j
(594) \nIÂqÅ YÁ Ï¿[RÀ \Á qÄ q¿[À^;Â 4Å ÁARÀ ÂqÅ Ån¿ Á 8K¿[Å3¾ V: MrjqM¢R1sPjf`o`RVa[MPRp4\^ja acMP\ a acMrRZkac\ne Q]PZkalKRpJeq\^ja[f^sP\^bIjf0f0e[R+oI a[M\a[eq\^j
(595) R jf0sPZkZka[R
(596) s3a a[MX:» Rp]PZkRp\ Kf N®«0R¤o`R+Z[j
(597) e[m`acf0sJf^ea[MrRpmrecf^PR1Q MPjqM j
(598) \nsKRVecR
(599) \ a[R+oacfR
(600) «{srf sImrecf^m` R1ekacR+Z/b \0©Ia[MrR°Q\nacjqMrsr~SmKf0b3acf^mAR°fn \rmP\^ekacacR°~^eq\nmrMZ R1sPjf`o`R+o {b ZkQSmrR Pf R 3]P\na[f^sPZ1 K©R 3]r)@mre[f0P\nPa'bQ\na[ecj
(601) R1Z \necR°jf0s30R_ j
(602) f^QPsP\na[f^sfnmKR1e[Q]`ac\na[f^s QS\na[ecj
(603) R1Z1. r_rtsuqvwxv.
(604) T6|.
(605) ¿¿ ¿[À.
(606) ¿
(607) ÁPÁ. e ; [ . 6ÀnÁAÅ. U. ¤Zf^]Pemrecf^rR
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(609) alfnf0\0o³psPo`R1mKR1sPo`R
(610) s3aDZ[RaqZ°¥JRZk]rKZkR
(611) acZNfn@o`mP\ acMPZNf0\^orsr~SR+\^jqM\^ecjVR
(612) _`\0jacb f0sPjR6©@½ eqZkaRV r_ \nssPorR
(613) mAR
(614) sPo`R1s3aDZkR
(615) a H \nsPoRpjf0sPZ[o`R1e:acMrRVJf^f sr~S\n]`_`\necb Pf mPe[f0rR1 Q b I½rf0e R1\^jqM¢o`mP\ ac& M ?
(616) E R°\^oro \ KÈ 6 È A5?
(617) El I½rf0elR1\0jqM¢j
(618) f^]rmPR°f^o`mP\na[MP Z ? E B ?
(619) YE Zk]KjqM¢a[MP\na ? E!= Y \B$\\nsP) o ?
(620) YE = Y nB7G \<R \^oro¢\^s \^ecOj ?A<? EBA5? qEE[ RCo`R PsPR\ Pf J]rsPjacf0s f^sa[MPR\neqj
(621) ZS MrjqM8sPo`]Pj
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