Choosability of bipartite graphs with maximum degree $Delta$
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Choosability of bipartite graphs with maximum degree Stéphane Bessy — Frédéric Havet — Jérôme Palaysi. N° 4522 Juillet 2002 THÈME 1. ISSN 0249-6399. apport de recherche.
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(111) !lÉxn»qu]b/qtйx*|xnbjxn» ,#ß , ¹l , *wkb6j}<b|qkÀnbsnªÉ«~`{¼D? ô`¦uï A |xJ|²pbj^Ð/qk]©¼ ? ïp¦èAÉì¾wkb6j}ªÉ¼ ? ï¦uå A íÉxn©£¤¢ , ¥Yó
(112) ¦ õ ì¾wkb6j}4ª£¤¢ , ¥!*ó
(113) ¦ õíªørxqkbqk]lqfqk]b@nwl]jTS ²jf¨`lwkqkqkbcl`©qk]`lqcln`b@?qk]b julacb`lnw}q~xn»qk]b¨``lnw}qu°quxJª xnwYlQs¤qtйx8qubnbwuCj *¡ Ë ß`bqY¼ ? ¦ A4¨<bqk]`b²j}qYlnjuj}n`a@b8q xJ¨pquln`b»ÅwkxJa¼ ? ô`¦uï A`¨Qslns8l*<bwua!mpqlquxJ F
(114) xJqu]br|xnxnmwjÉóJèn¦ ¦uïpõ^Ð/]`|]¤julquj}ð`bj F¢Mô¥%H ¡ ¿l F¢ÍïQ¥%¡ ª (. . +5. 5. 5. 5. ?5. 5. (. M1 L M(5,1) {2,1,5}. {4,1,5}. a1 s {4}. b1. t. u. {4,3,5}. v. {1,3,2}. w. {5,2,1}. {5}. b2. a2 {4,5,2}. {1,5,2}. M2 L M(5,2). nm`wkb þH\^]bJwuln]Sùln`+quj~²jtqrlJjkjknacb8q/¼ ? ô ¦uï A[ª (. . * , # Û!´¶ IY¡Ë e%3S(¯²±¶<Û³ ¼ ? u¦ A[Ï}ÙÛÛ k´8ܸ 4 m`x8j}b^qu]`lqhSÆlJpac°qjFl*¼ ? ô¦uï AÁÂ|xnxnm`wkfÃ@ªb=Éxnxnm`wHj%»ÅxJwk¨²pbcxncÀnbw}qu|bj
(115) ß
(116) ln` ^l`|xJxJmwY@²j^»Åxnwu¨²pb{xnÀnbw}qubo½<ª¹ÞÍ»FÃ@¢M½`¥¿¡Äènß`qk]bOÃ@¢ ¥Y¡ÄË èl©Ã@¢¾Ê ¥Y¡ÄË ènª z{xnwubxÀnbwßnÃ@¢ ¥%|l`xq¿¨<bYpßJxqu]bwuÐ/jkbrÃ@¢
(117) u¥H¡ýln`Ã@¢
(118) ¥H¡ñåln`@Ê|lncxq¿¨<br|xnxnmwubت \^]`bVÃ@¢ n¥Y¡Ñåln`ÈÃ@¢
(119) ¥f¡Õï¨mpqqu]j|xJJquwulJp²|qujvFwkxJx8j}qkxnÈpª =ÉxJ`jkb 8mb8qusJß4½Oals `xq+¨<b?|xnxnmwub dnªD«r`lxnJxnm`jksnß¹¨8sjks8acacbquwksJß/½×|lxnq+¨<b?|xJxJmwubpª xß S j+xq ¼ D? ô`¦ï AÁÍ|xnxnmwl¨bnª " bq #l`
(120) ¼ C? ô`¦ï A¨<b*qk]bJwuln]ln`
(121) °qj^²jtqrlJjkjknacb8q~pb|qkb6 %nmwub*`ªH\^]bÀJbwkqk²|bj
(122) Qü~ln` ü/lwub/²pb8qkð`b6cÐ/°qu]cqk]b~ÀJbwkqk²|b6j
(123) !l ¿xn»Ul!|xnQsxn » ,#ß ,ÈüªH«r`¼ C? ô`¦ï A|xn`|²pbj Ð/qk]X¼ ? åp¦Bè A^ì¾wkb6j}4ª¼ ? ý`¦è AÅí/xJX£¤¢ ,Èü!¥ *ó
(124) Qü¦ üõ{ìÅwubjkª!£W¢ S¬ ¥ fó J¦ õn¥uª!\^]b@nwl] Ñj ¨``lnw}qu°qubln` cln` #b!?p Ubwub8qflwkqYx»¿qk]b@¨`lwkqkqkxnª `xnKw !¡ Ë `ßUbq¼ C? ¦ A%¨<b@qk]b ²jtq*lnjuj}nacbJq~xJ¨pquln`b»ÅwuxnaƼ É¢Mô`¦uïp¥¿¨Qs
(125) lnsQl¤<bwua!mpqlqkxn F¬xJ{qk]`b!JqubJbwj/Ð/]²|] julquj}ð`b6>j F¢Mô¥H¡ ¿ln` F¢ÍïQ¥¡ ª 4 * , # Û !´¶ IY ¡ Ë e% Ư²±¶<Û³%¼ C? ¦ A[Ï}ÙÛÛ k´8Ü,¸ 4. . ". . 5. (. (. ". . =,?I =BA.
(126) ÚpÛÛ±´8ܯ °¯ ³ Û Ü¯ p´,³M¯ ³Â¸*µ,u´ `ÚJ± . . . . M3 a3=w. L M(2,1). {4,1,2}. b3. N. y {4}. x. {5,2,1}. L N (3,1). {5}. s. %nmwub*`þF\^]b*Jwuln] l`quj/j}qrlnjujkJacb8q^¼C? ô¦uï A[ª {4,3,2}. (. bq!ÃѨ<b¤lȼ ? ô`¦ï AÁÍ|xnxnmwu`+x» cª\^]bVÃ@¢ ¥¡#Ë ôß Ã@¢
(127) ü ¥¡#Ë ô{l`¬Ã@¢ ü ¥c¡ Ë ïpª `mwkqk]bwkacxnwubÉÃ@¢ü¥%¡ åpª rqk]bwkÐ/²jkbnßÃ@¢ü¥¡ è^jkxfÃ@¢
(128) Qü¥¡ñårÐ/]²|]!|xJJquwulJp²|quj vHwuxn<xJjk°quxJpª rb`|bOÃ@¢ ¥¡-è©ln`Ã@¢ n¥¤¡1ý`ª÷\^]²j¤|xJ8qkwlnp²|qj@vFwkxJx8j}qkxnñpª x ®²jxnq+¼C? ô`¦uï AÁ |xnxnmwl¨`bJª " bq 1¨<b{qk]`b©nwl]xJ¨pqulnb6Ö»ÅwkxJa qtйxV|xJb6j l` {¨Qs×pb8qk»Ås8Èqu]b©ÀJbwkqk²|bj |xnwubjkxJ`pquIx Ul`m X¤ln`¤lJpln¤bpJbr¨<bqtйbb¤qk]`baª bb %Jmwkb*pª«~`bq¹¼<? ô Aبb qu]bj}qflJjkjknacb8qr|xn`|²p¤Ð/qk]©¼ C?°èn¦kô AÉì¾wkb6j}4ª~¼ C? åp¦}ô A í¹xJO£¢ ¥ X¬ìÅwubjkª~£¢ n¥ [Xpí ln`
(129) ¼ <? ô AÍ¢ X<¥%¡óJèn¦åp¦kôõJª 4 (. 7
(130). .. 75. (. 5. P1 L P (1,4). y. x. {4,1,2}. {4}. P2 L P (2,4). %JmwubþF\^]b*nwl]l`
(131) quj^j}qrlnjuj}n`a@b8q/¼ ? ô A[ª (. * , ö¯²±¶UÛ³¼ ? ô AÍÏ}ÙÛÛ k´Jܸ 4 =Éb6lwusJß Õj*xnq¼<? ô AÁÍ|xnxnmwl¨`bJª!ÞÂ`pbbXOals{xnq¨b|xJxJmwub©¨<b|lm`jkbcx»¹qk]b b6pnb@óX<¦ Uõl+¨Qs+vHwuxn<xJjkqkxn
(132) @qr|ln`xq~¨<b|xnxnm`wkb6©è*xnwrå¨<b|lnm`jkbfxn» fl nª " bqr B l`
(133) ¼ B ¨<bfqk]bJwuln]pb²|qkb6 %nmwub þ -FlJ|] ßØè*î Éî÷
(134) è `ßpj/l@|xJQsx» l ¼È|xn`|²pb6jFÐ/qk]¤¼ /¢¾ô¥ xnblJ|] nU ª Y¨QÀ8xnmj}snßQ B j¿¨`lwkqkqkbrln`c]`lJjHalopama pbJwkbb ª 4 * , ö¯²±¶UÛ³ o¼ BÏ}ÙÛÛ k´8ÜL¸ %F±Û¯ ³¹¯²±¶<Û³ïÏ}ÙÚpÛÛ±´8Ü,¸ 4. . 75. (.
(135). . @. )=?>4@#7#7.
(136) d6. r¸±± ´¸³~´´ ±¯ . Q. Q. 3 LQ(4) {2,3,4}. Q. 2 LQ(4). . . {1,2,3}. {1,3,4}. x 1. . x5. . . . . z1 x 10. z2 x
(137)
(138) 9. {1,3,4}. Q. Q. 1 LQ(4). Q. 5 LQ(4). x3 {2,3,4} x4. {1,2,4}. x2. Q. 4 LQ(4). 6 LQ(4). . {1,3,4}. x6. {2,3,4}. {1,2,3}. {1,2,4}. x 8. . {2,3,4}. Q. 10 LQ(4). . . Q. 7 LQ(4). x7. {1,3,4}. Q. 9 LQ(4). 8 LQ(4). %Jmwub `þF\^]b*nwl]{ Bl+quj/²jtqrlJjkjkJacb8q/¼hBª (. 4 m`x8j}b@qu]`lq B lnpacquj*l
(139) ¼ B ÁÂ|xJxJmwu
(140) Ã@ª"-HÀJbwus^U ßFè¤î *î#è ßU²j*xq|xJxJmwkb6 ¨Qs?vFwkxJx8j}qkxnÈpªÞÍ»~Ã@¢ ¥r¡ ýßqu]bºÃ@¢RU ¥Y¡ ý{l`ÈÃ@¢RUün¥r¡Ñål`Xйbc|lXxq!|xnxnmw U ªñ«r`lxnJxnm`jksºÃ@¢ ¥¡ ýXbln`j@qkxÈl¬|xn8quwulJp|qkxnª×ÞÍ»Ã@¢ ¥¤¡ èqu]bñÃ@¢ ¥¡1åpß^jkx Ã@R¢ UJ¥¡ Ã@¢ U;B¥¡ ý¬Ð/]|]²j+lÈ|xn8qkwln|qkxnª «~`»!Ã@¢ ¥+¡ è{qk]b÷Ã@¢ ¥+¡ åXqk]Qm`j Ã@R¢ Un¥%¡÷Ã@R¢ U ¥H¡ýcÐ/]²|]²j/l|xJ8qkwlnp²|quxJª " ¬p!
(141) 8¶÷³MÚQ¸ uÛÛ ?Û
(142) kÛ Û±¯ ³[¯¾Û# ¶ "%³MÚQ¸O¶<Û¶ ÙÛÛ k´Jܸ? ¯²±³¤´±±¯°µ¶·c¸¶`³ Q±¸±%$h¸ ¯ r$¸ k¸¶³~ÙuÛÛ ± %ØÚ8¯²±!¯²±³MÚQ¸·¯ ¶¯ · p·ö¶ p·¤Üu¸ !Û ÙÛÛ ,Û @´
(143) ÙuÛ p¶`³Â¸ u¸ Q´· ,¸ T8¶ J¸¸ +¯ !´ °¯²±³/´±±¯°µn¶`·¸¶³ ¼ Q±¸±Û¶` Û !ÙÛÛ ³¾Úp¸¶?¯ ³É¯²±!´ ¿´ ±r¼~Ï}ÙÛÛ k´8ܸ´±Û8ܱ$¸ ¸ +¯ '¶ &)(+*L ïRÏ k ¯²± T¹Ï}ÙÛ· ¸³Â,¸ 4 b-q ,÷¨<brqu]bYjkm¨Jwuln]x» B pm`|b6¤¨Qsqk]bfÀnbw}qu|bjFxn»4£¢[ B ¥ É¢[£+¢ 6¥ ¿,ó U 6õJ¥%l bq/¼ .>?°è A¨bqk]`b!j}q*lJjkjknacb8qYbð`b6©¨Qs{/¼ .>? ô AÍ¢¾½¥^¡¼ B ¢¾½¥^°»¿½OÇÈ£¤0¢ ,¥kª xJwflnQs
(144) 8qkbnbw ßpbqr-¼ . ? A4¨<b*qk]bj}qrlnjujkJacb8q¹xJ¨pquln`b+»ÅwkxJa$/¼ .>? ô A¨Qs+lns8¤l@<bwua!mpqlqkxn F?xn
(145) qk]b 8qkbnbwjÐ/]²|]Èjklqk²jtðb@j F¢Mô¥*¡ ª wuxna qu]b+`wkxQx»^x»~vHwuxn<xJjkqkxn ß%ÐÉbpbpm|bqu]`lq!bÀnbwks -¼ .>? ô AÁÂ|xJxJmwkcÃDxn1» ,jklqk²jtð`b6j~Ã@¢ U ¥%¡ñô`ª¿\^]Qm`j¹»ÅxJw~lQs¤8qkbJbw ßbÀnbwus¤/¼ .>? AÁÂ|xJxJmwkà xn2» , jklqk²jtð`b6jfÃ@R¢ U ¥¹¡ ª bqc¢[ @¦u¼/¥¹¨bcl©`j}quln`|bx»Fqu]bT« ? =¹v%ªU«rOb Jm`Àlb8qf`j}quln`|b ¢Í *Î ÎM¦u¼¿Î ξ¥¹als¨bcxJ¨pqulnb6»Åxnwfqu]bcïjÁ ? =¹v/þ xnwfblJ|]©Ànbw}qub^o UVǺ£c¢Í ¥/Ð/]xJjkb²jtq¼~R¢ U¥/]`lnj |lwp`l°qtsOß4jkbq¼ Î Î ¢ U¥Y¡ ¼r¢ U4 ¥ ¬ó LJõ@»Åxnwjkxnacb LÈÇºä ¼~¢M½`¥ußytxn' U¬qkx
(146) qk]`bÀnbwkqkbo©½©x»^l|xnQs , @x1» ,Dl`
(147) jkbq~¼¿Î ÎM¢¾Ê ¥%¡¼ . ? L A[¢MÊ4¥»Åxnw/Ê{Ç©£¤0¢ , p¥kªHÞÍq/²j^blJj}scquxc|]`b| cqk]lqY *Î ÎØ]`lJj¹bnwubb*lq acx8jtq^ð`ÀJbnª " 9 3
(148) O/Õ ?!, H+.
(149) 5 4 ý76 9 0 8 \^]`bJwuln]× B ]`lJj n8©Ànbwkqk²|b6jßÉj}x?ûü¢[ïp¥¤; î :nô8åª ~xйbÀnbwßHxJb|lÖnbqlXjkn]8qksȨ<bq}qubw ¨<xnm``Øþ 5. 5. 5. 5. 5. . . !. . . . . . . . ". !. . . . . . . '. . . '. 5. ,5. (. (. ,5. 5. ,(. . .. =,?I =BA.
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