the list coloring conjecture

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Department of Computer Science | Institute of Theoretical Computer Science | CADMO Theory of Combinatorial Algorithms Prof. Emo Welzl and Prof. Bernd Gärtner People Personnel Guests Alumni Reports Current Report Report for 2017 Report for 2016 Previous Reports Research Grants Dissertations Master/Bachelor Theses Publications 2018 Publications up to 2017 Mittagsseminar Teaching Courses Theses + Topics Events Workshops Current Previous Social Activities SOLA GREDA GRAPE ??>þÿ ¹þÿÿÿþÿÿÿµ¶·¸ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ Search ethz.ch Search Person

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THE LIST COLORING CONJECTURE. nicla bernasconi. topics. Introduction – The LCC Kernels and choosability Proof of the bipartite LCC. Introduction and the list coloring conjecture. vertex coloring. k-coloring of a graph G: labelling f: V(G)  S with |S|=k. The labels are called colors - PowerPoint PPT Presentation

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Page 1: THE LIST COLORING CONJECTURE

Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms Prof. Emo Welzl and Prof. Bernd Gärtner

• People • Personnel • Guests • Alumni

• Reports • Current Report • Report for 2017 • Report for 2016 • Previous Reports

• Research • Grants • Dissertations • Master/Bachelor Theses • Publications 2018 • Publications up to 2017

• Mittagsseminar • Teaching

• Courses • Theses + Topics

• Events • Workshops

• Current • Previous

• Social Activities • SOLA • GREDA • GRAPE

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?=???¨THE LIST COLORING CONJECTURE¡ª¨nicla bernasconi¡ª?¨topics¡ª� � � �  Introduction The LCC �Kernels and choosability Proof of the bipartite LCC¡KKª �

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? ªª?¨vertex coloring ¡"ª� � � � k-coloring of a graph G: labelling f: V(G) ? S with |S|=k. The labels are called colors A k-coloring is called proper if adjacent vertices have different colors G is k-colorable if it has a proper k-coloring Chromatic number: c?G):= min{k|G is k-colorable}¡4?Z Z !%1 ª^ ?¨example¡ª?� �� ��� �

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¨edge coloring ¡" ª" � � k-edge-coloring of a graph G: labelling f: E(G) ? S with |S|=k. A k-edge-coloring is called proper if incident edges have different colors G is k-edge-colorable if it has a proper k-edge-coloring Chromatic index: c? (G):= min{k|G is k-edge-colorable} ¡^MX&Xÿþ, .$ ªL�� � �� � �� �

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?C ¨example¡ªª? ¨ line graph¡ ª"¨zThe line graph L(G) of a graph G is the graph with vertex set � � � �V(L(G))=E(G) and edge set E(L(G))={{e,f}| e incident with f}¡8{hª¾ ?¨example¡ªª?D!¨example¡ªª?¨ list � � � � �coloring¡ª"¨List coloring is a more general concept of coloring a graph G is n-choosable if, given any set� � of n colors for each vertex, we can find a proper coloring ¡ ÿþ3 N$$ª� � �

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? ¨example¡ªª? ¨ choice number¡ª � � � � Choice number: ch(G):= min{n|G is n-choosable} The list chromatic index of H is the choice number of L(H) Clearly: ch(G) ³?c?G) ¡8!; -ª? �����

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?F#¨example¡ªª?� �

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¨list coloring conjecture ¡ª&� � Pch(G) = c?G) whenever G is a line graph ¡L)ªt��

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?ªª?¨ definitions¡� � �

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ª � Consider a digraph D=(V,E) Notation: u ? v means that (u,v) ? E:¡¬B ªN � ����

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?¨kernel¡ª� � *A kernel of D is an independent set K ? V s.t. "??V\K there exists an u ? K with v ? u A kernel of S ? V is a kernel of the subdigraph induced by S¡$ÿþ *ª?� � ��� �� �� �

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?H%¨example¡ªª?G$¨(f:g)-choosable¡ª& � � � � Consider two functions f,g: V ? ? G is (f:g)-choosable if, given �any sets Av of colors with f(v)=|Av|, we can choose subsets Bv ? Av with g(v) =|Bv| such that Bu '?Bv = ? whenever {u,v} ? E ¡~?Z ????? ? ? ?ª8��� � �� � �� ��� ���� �� �� � ��� �����

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?¨lemma 1¡ª&� � Bondy, Boppana and Siegel Let D be a digraph in which every induced subdigraph has a �kernel f,g: V(D) ? ? are so, that f(v) ³? whenever g(v)>0 Then D is (f:g)-choosable¡?ZIdZI??dZC??dZ0 ���EªF�� �

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?ª� � Given: sets Av with |Av|=f(v) Goal: find Bv with |Bv|=g(v) and Bu '?Bv = ? whenever u,v adjacent Define W := {v?V| g(v) > 0} Choose color ¡¶ " Z ? ? ? ? ? ? ª¬?�� �� � ��� ��� �

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?&ªª?*ª� � � �  Induction hypothesis ? G is (f :g )-choosable This means: $?v ?Av\{c} with |Bv |= g (v), s.t. Bv '?v = ? if u,v adjacent Define ¡

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z¶ " ·

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" ¶ " ·

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" ? ? ?? ? ªha� � � ��� �� �� ��� �� �� ��� �

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?:¨(m:n)-choosable¡ª* ¨bG is called (m:n)-choosable if it is (f:g)-choosable for the constant functions � �f(v)=m and g(v)=n ¡<cZ

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Hªa

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?J'¨ corollary 1¡�

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ª* ¨¦Let D be a digraph in which the maximum outdegree is n-1 every induced subdigraph has a kernel �Then D is (kn:k)-choosable for every k. In particular D is n-choosable.¡jIdC??dH??dBIªB �

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?<ªª?=¨ definitions¡� � �

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ª � An orientation of a graph G is any digraph having G as underying graph Let K be a set of vertices in a �digraph. Then K absorbs a vertex v if N[v]'? ¹??. K absorbs a set S if K absorbs every vertex of S¡<?Z ÿþ, �� ª? ?>¨the bipartite LCC¡ª* ª??ª���� �� ����� � � � � � ¤Proof: Let V := V(G) = E(H) Define Ax:={v?V| v is incident with x, x?V(H)}. Ax is called row if x?X, column if x?Y If v is a vertex of G, then R(v) is the row and C(v) the column containing v Take a proper coloring f: V ? {1,...,n} of V Define an orientation D of G in which u ? v if R(u)=R(v) and f(u)>f(v) or C(u)=C(v) and f(u)<f(v)¡^·0 " Z2¶0 " Z25·0 " Z2? ? 0%4ª )���� �� � ����� �

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k

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?@ª� � ?Check i) and ii) of Corollary 1 i): ? (since f is one-to-one on N[v]) ii): induction on |S| Given S?V, show the existence of a kernel in S Define T := {v?S| f(v)<f(u) whenever v¹? ? R(v)'?} T absorbs S If T is independent, then it s a kernel ?¡|¶ " 2-· " 26¶ " 2$ =0 6 ªp� �� � �� ������ ����� � �� �

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Page 47: THE LIST COLORING CONJECTURE

?Aª� � ?Assume T is not independent, then it has two elements in the same column, say v1,v2 ? T with C(v1)=C(v2)=:C, f(v1)<f(v2) Choose v0 ? C'? s.t. f(v0)<f(u) whenever v0 ¹?u?C'? Induction hypothesis ? S\{v0} has a kernel K, and K absorbs v2, this means N[v2]'? ¹?? N[v0]'? ?N[v2]'? ? K absorbs v0 ? K is �a kernel of S ¡ff¶ " 2N?? ?? ?? ? ?? ? ? ? ? ? ��� �� �� �� �� �� �� �� ��� ��� ��� � ���� �� �� � �� ���� �� �� ����� �� ����� �� �

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§*eN5l @¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿??OV?O? ? ? ?? ?"?¿? ?? ?? ? ?BlC?DE?F?d?@¿ÿ ?l??ll?l?� �� � � � � � � ���������� ������ ����� �@¬¬¬¬¬¬`#"?¿`¿? ?? ? ??? ?? ?"?¿????2 ?? # ?¿???d @¿ÿ?¿ÿ #"?¿`¿?f?z)?2 ?? ?¿??? @¿ÿ?¿ÿ #"?� � � � � � � � � �� � � �������� � �� ��� ��������� �¿`¿?R?2 ?? ? ?¿???d @¿ÿ?¿ÿ #"?¿`¿?? ??2 ? ? ? ?¿??? @¿ÿ?¿ÿ #"?¿`¿??8???2 ? ? ? ?¿???d @¿ÿ?¿ÿ �� � ��� � � � �� ��� � � �� ��� � �#"?¿`¿?R?½? ? ? ? ?B~C¡DE?F???? @¿ÿ ?x� �� ����� ����������� ����� �

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Page 129: THE LIST COLORING CONJECTURE

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Page 130: THE LIST COLORING CONJECTURE

?$ITyr£?º?¢~s`CB 0� � � �

Page 131: THE LIST COLORING CONJECTURE

N04@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿? ??¨? ?? ? ?BzC?DE?F???yÿd? @¿ÿ 44??;®_`?� ����� ���� � ������ ����� � ��<$l?a$©<?`?®]?cWJ?zª????3ªWztJzt?W¨-x?Tµ0m?f?0TMx#¨� �

Page 132: THE LIST COLORING CONJECTURE

?J#zGª}?§?kª;z#Jjl@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?�¿`¿?^

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??? ?? ? ?BC?DE?F???¦ÿd? @¿ÿ ?,?0?Sl¹H?*IC� ����� ���� � ������ ����� � �� ��

Page 134: THE LIST COLORING CONJECTURE

?$³BwfH??,\B¼k??S?0b,,:<@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?"?¹ ?? ?? ? ?BkC?� � � � �DE?F???d? @¿ÿ ?YTSA®$???5?S®ekTe*YAS*YTYT"$@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?« ??? ?? ?����� ����� ������ ����� � � � � ?BCCDE?F?¿ÿ ?Uª?*<C*?°� ����� �

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Page 136: THE LIST COLORING CONJECTURE

?$nNz`?6z*,@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?eW ?? ?? ? ?B«C?DE?F?¿ÿ ?+¿CtH«V<+x �� � ����� � � �¿=m1V?z?V7m%=*,@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?K ??? ?? ? ?BhC3DE?F???d? @¿ÿ ?� � � ����� ����� ������ ����� �hma1O?1??lB\� �

Page 137: THE LIST COLORING CONJECTURE

C*N½r?¢?+=7Cm=£+??3�

Page 138: THE LIST COLORING CONJECTURE

37O?a£hmhm:<@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿? §??? ? ? ? ?B6C©DE?F???� � ����� �����d? @¿ÿ ?©6£*yP??´*x¶H0¤������ ����� � �

Page 139: THE LIST COLORING CONJECTURE

Page 140: THE LIST COLORING CONJECTURE

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Page 141: THE LIST COLORING CONJECTURE

¤**ªT???,ÿV©©8<@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿? /D ?? ?!? ? ?BbC?DE?F???�� � � � ���� �����d? @¿ÿ ???7ºP\Zb$bJJ$CZ7º?? @¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿???J ?? ?"? ? ?B?CDE?F?¿ÿ ?»s71[?������ ����� � � � � � ����� � �º?$T0?� �

Page 142: THE LIST COLORING CONJECTURE

?g$N?~´I?%1s»?!+QIaCQ+!?»»:<@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿??Y???? ? � � � �gU ?#? ?"?¿? gU? ?$? ?B±C?DE?F???? @¿ÿ JJ?\?r³le¢0?� � � ���� � ���������� � ���� �

Page 143: THE LIST COLORING CONJECTURE

&Jn �

Page 144: THE LIST COLORING CONJECTURE

µ0?e9³or½??\???E½ªoL9?¥µ±±n±J¥&?L¢lª<E?\\\??½~«?c-S?0µ*$n*J0,S?®?~T~6?\?96T?� � � � � � � � � �~.®d?,nµd?.-?c9«?½\?\?� � � �� ��@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?hJ %? ?%? ? ?B C?DE?F???? @¿ÿ $$?$JBl?� ����� ����������� ����� � � ?fO<´ �

Page 145: THE LIST COLORING CONJECTURE

®I0?ZZ?*J??*!NQr{¡???C???{lQH'0??�� � � �� ��JL@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿? & @? ?&? ? ?BaC?DE?�F???? @¿ÿ ?����� ����������� ����� �

Page 146: THE LIST COLORING CONJECTURE

q?$0xN?r7?a?a?1¨?~´Zl<$?k�

Page 147: THE LIST COLORING CONJECTURE

24@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?¸ ? ?'? ? ?BHC6DE?F???d? @¿ÿ ?� ����� ����� ������ ����� �H$6$66*HHHH@¬¬¬¬¬¬¬¬¬`#"?¿`¿?~§??? ?(? ? ?B?ClDE?F???d? @¿ÿ ?+Z?ZN0B`?l+lfTg<??� ����� ����� ������ ����� � � � �g*BT+Z+Z&(@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿??ZU? ?)? ? ?BCUDE?F???d? @¿ÿ ?wrq� � � � ����� ����� ������ ����� � -Y?/%UUMq? �r<wYq<wrwr"$@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿??Fg? ?*? ? ?BSCZDE?F???d? @¿ÿ � � � � � ���� � ���� � ����� � ���� � ?;ZSTG<5*#0;Z;Z@¬¬¬¬¬¬¬¬¬`#"?¿`¿??(? ?+? ? ?B?C¯DE?F?¿ÿ ???µ?¯%yIVg ?¨g£%?� � ����� � � � � � � � �wgTI0%?®0TfwH� 0?%g

Page 148: THE LIST COLORING CONJECTURE

¢?0DBJ0?

Page 149: THE LIST COLORING CONJECTURE

¢g?¡$kB<`®?+<Oks¡?©g¯®©?2bsO¯+???????¯?µ?µ?µ?� � � � � � ��@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?]?*? ?,? ? ?BCDE?F???¦ÿ? @¿ÿ ::?hh?<?T'rE]´i?oi7Kg'?»¶?n???s????»fBg$7$?� � � � ���� � ��� ������� ����� � � � � �0ºBZ~Br$´?7$mN????m? z??? ??im7?u®]9fB?� � �� � � �� �*hhvx@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿??¶ ?? ?-? ? ?BmCBDE?F???? @¿ÿ ??+gm� � ����� ����������� ����� �

Page 150: THE LIST COLORING CONJECTURE

1?®r

Page 151: THE LIST COLORING CONJECTURE

*B60r®?? $@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿??¤ A?? ?.? ? ?BBC0DE?F???? @¿ÿ ?B0� ����� ����������� ����� �

Page 152: THE LIST COLORING CONJECTURE

0*BB@¬¬¬¬¬¬¬¬`#"?¿`¿?o?±?2 ?/? ? ?¿???¦ÿ @¿ÿ?¿ÿ #"?¿`¿???W?2 ?0? ? ?¿??? @¿ÿ?¿ÿ #"?� ��� � � � � � � �� � �¿`¿??.��  =?2 ?1? ? ?¿???d @¿ÿ?¿ÿ #"?¿`¿?B+?2 ?2? ? ?¿??? @¿ÿ?¿ÿ #"?¿`¿?.^X?2 ?3? ? ?¿???d ��� � � � �� � ��� � � �� ��� � �@¿ÿ?¿ÿ #"?¿`¿?Rr0??2 ?4? ? ?¿??? @¿ÿ?¿ÿ #"?¿`¿???? ?� �� ��� � � �� ��  ?I? ?5? ?"?¿?�  ?I?? ?6? ? ?B~C`DE?F?d?���������� ������ �����@¿ÿ ??`ZSB#$*MHZ?`Z7T`B~*x*ZB1NZ?`?`*,@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?� � � �  r

Page 153: THE LIST COLORING CONJECTURE

?? ?7? ? ?BC6DE?F?d?@¿ÿ ?®???®xN���������� ������ ����� �

Page 154: THE LIST COLORING CONJECTURE

$06$$Nx®®"$@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿????? ?8? ? ?B<CDE?F?d?@¿ÿ ?� � ���������� ������ ����� �6Z0~6~<Z6B0***0B6Z6Z"$@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿? ?I�� � �

Page 155: THE LIST COLORING CONJECTURE

? ?9? ? ?B?CDE?F?d?@¿ÿ ���������� ������ ����� �

Page 156: THE LIST COLORING CONJECTURE

?r

Page 157: THE LIST COLORING CONJECTURE

H6

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r?�

Page 159: THE LIST COLORING CONJECTURE

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Page 160: THE LIST COLORING CONJECTURE

r

Page 161: THE LIST COLORING CONJECTURE

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Page 162: THE LIST COLORING CONJECTURE

¿?? ?:? ? ?B¡CºDE?F??@¿ÿ ? r`N5*e¡_���������������� ����� � �

Page 163: THE LIST COLORING CONJECTURE

/$ HZr ¨)ºAº)¨ r r,0@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?¸?Y� �� � �

Page 164: THE LIST COLORING CONJECTURE

? ?;? ? ?B¹C?DE?F??@¿ÿ ?B���������������� ����� �

Page 165: THE LIST COLORING CONJECTURE

w$H¡Z§r¡¢w´Z?`??¡¨³¹r³Z� �� � � B0H24@¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬`#"?¿`¿?q?+´� � �

Page 166: THE LIST COLORING CONJECTURE

? ?? ? ?¿?¿ÿ?¿ÿ #"?¿`¿? ��� � ��

Page 167: THE LIST COLORING CONJECTURE

?�

Page 168: THE LIST COLORING CONJECTURE

?2 ??? ? ?¿?d¿ÿ?¿ÿ #"?¿`¿??���� � �� �

Page 169: THE LIST COLORING CONJECTURE

?2 ?@? ? ?¿?¿ÿ?¿ÿ #"?¿`¿?7%��� � ��

Page 170: THE LIST COLORING CONJECTURE

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Page 171: THE LIST COLORING CONJECTURE

?2 ?A? ? ?¿?d¿ÿ?¿ÿ #"?¿`¿?R5���� � � �

Page 172: THE LIST COLORING CONJECTURE

d�

Page 173: THE LIST COLORING CONJECTURE

? ?B? £ ??[¿¿?ÿ "?¿?*°?p??[ ?¨ Click to edit Master title style¢!ª !? ?C? ??¿?ÿ "?¿? ` ???[ ?¨#Click to �� � �� � � ��� �� � �edit Master subtitle style¢$ª $? ?D? ??¿?ÿ #"?¿`¿?` `??[ ?� �� ��� � � *¡?? ?E? ?,?¿?ÿ #"?¿`¿?`°??? [ ?� �� ��� � � *¡?? ?F? ??[¿?ÿ #"?¿`¿?` `??[ ?� �� ��� � � *¡?? ??

Page 174: THE LIST COLORING CONJECTURE

s ? �? ?h¿ÿ ?"?¿`? h®ÿÿÿ??ÿ?ÿ¯?º___PPT10i������ ��� � � ���� � �  ?+D?'?¹ÿÿÿÿ=?@B? +?? 0� �

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??°?,?? ? ?,? ?, ?(?¿?ÿ ?P ?? ? ?� �� � �� � *¡

Page 176: THE LIST COLORING CONJECTURE

?ª? ?, ???¿?ÿ ? ? ??? ?� �� �� � � *¡

Page 177: THE LIST COLORING CONJECTURE

?ª? ?, c ?¿ÿ ??°? ???? ?, ?X?¿?ÿ ?° @��� � �� � ��  ???? ?¨uFare clic per modificare gli stili del testo dello schema �Secondo livello Terzo livello Quarto livello Quinto livello¢:ªv? ?, ?DT?¿?ÿ ?`P?? ? ?� �� ��� � � *¡

Page 178: THE LIST COLORING CONJECTURE

?ª? ?, ?´X?¿?ÿ ?` ???? ?� �� ��� � � � *¡

Page 179: THE LIST COLORING CONJECTURE

?ª? ?,

Page 180: THE LIST COLORING CONJECTURE

?� ����????h¿ÿ ?? ÿÿÿ33??ÿ²²²��� ��� �

Page 181: THE LIST COLORING CONJECTURE

?0??? ? ?? ? ??[¿ÿB??*°?p?$?D 0?[ ?? ? c ??[¿ÿC??P ` 0� �� ��� � ��

Page 182: THE LIST COLORING CONJECTURE

?$? 0?[ ?? ?�

Page 183: THE LIST COLORING CONJECTURE

?� ���?h¿ÿ ?? ÿÿÿ»?33?¤º___PPT10|??<7? ?+(R!?D?'?¹ÿÿÿÿ=?@B? D?'?ÿÿÿÿ=?@B?A??%?(? ?B������� ���� � �� � �style.visibility?B?style.rotation?B?ppt_w?B?ppt_w?B?ppt_h?B?ppt_x?B?ppt_y?B?ppt_y?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility? s ?D¿??|ÿ?0�

Page 184: THE LIST COLORING CONJECTURE

??p ?$?@ 0?B ??? s ?D¿??|ÿ?° ??$?@ 0?? ?¢?F ?Y???¢F ?$?Dÿÿÿ 0?B ?@? s ?D¿?333?|ÿ???µ® ?� � � �B ?A? s ?D¿?ÿ?|ÿ??ª?� � ?$?@ 0?B ?B? s ?D¿?ÿÿ?|ÿ??"?% ?$?@ 0?B ?C? s ?D¿?ÿf?|ÿ???µp?$?@ 0?� � � �B ?D? s ?D¿?ÿÿ?|ÿ??`?% ?$?@ 0?B ?E? s ?D¿??|ÿ??`?6 ?$?@ 0?B ?F? s ?D¿?3f?|ÿ??% ???$?@ 0?B� � � � � ?G? s ?D¿?£ÿ?|ÿ??c ???$?@ 0? ?H? s ?¿?ÿ"?¿`?¢?*P?$?@ 0? ?I? s ?¿?ÿ"?¿`?¢*?$?@ 0? ?J? s ?¿?� � � ��� � ��� � � ���ÿ"?¿`?¢B*??$?@ 0? ?K? s ?¿?ÿ"?¿`?¢ * ?$?@� ��� �

Page 185: THE LIST COLORING CONJECTURE

0? ?L? s ?¿?ÿ"?¿`?¢¾ *F ?$?@ 0? ?M? s ?¿?ÿ"?¿`?P??$?@ 0? ?N? s ?¿?ÿ"?¿`??$?@ 0? ?O? c ?¿?� ��� � ��� � � ��� � � � ��ÿ"?¿`? ?$?@ 0? ?P? c ?¿?ÿ"?¿`??S ?$?D 0? ?Q? s ?ÿÿ¿?ÿ"?¿`??` ?P?$?D 0? ?R? s ?ÿÿ¿?ÿ"?¿`??� � � �� � � ��� � � ���®6P?$?D 0? ?S? s ?¿?ÿ"?¿`?`¡??$?D 0?? ??2¡� ���   ?Z??2?¡  ?$?Dÿÿÿ 0? ?T? s ?ÿ¿?ÿ"?¿`??2}� � ���

Page 186: THE LIST COLORING CONJECTURE

º?$?D 0? ?U? s ?3f¿?ÿ"?¿`?? }� ��� � �

Page 187: THE LIST COLORING CONJECTURE

?$?@ 0? ?V? s ?333¿?333ÿ"?¿`?` ?� ���   ?$?@ 0? ?W? s ?ÿf¿?ÿ"?¿`? ¡ ?$?@ 0? ?X? s ?ÿ¿?ÿ"?¿`?® 6� ��� � � ���   ?$?@ 0? ??

Page 188: THE LIST COLORING CONJECTURE

?� �?h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ¯? @@º___PPT10??S??$+?þD?'?¹ÿÿÿÿ=?@B? D??'?ÿÿÿÿ=?@B?A??%?(? ?B������� ��� � � � � � � � �style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility??�

Page 189: THE LIST COLORING CONJECTURE

¬¨ ?$?D 0?B ?+B s ?D¿??|ÿ?0� �

Page 190: THE LIST COLORING CONJECTURE

???$?@ 0?B ?,B s ?D¿??|ÿ?0� �

Page 191: THE LIST COLORING CONJECTURE

0° ??$?@ 0?B ?- s ?D¿??|ÿ??p°?$?@ 0?B ?/ ?D¿??|ÿ?° p ?p ?$?@ 0?B ?0 ?D¿??|ÿ?0� � � � � � � � � �

Page 192: THE LIST COLORING CONJECTURE

?° @ ?$?@ 0?B ?1B ?D¿??|ÿ?° ?pp ?$?@ 0?B ?2? ?D¿??|ÿ?`� � � � � � � �

Page 193: THE LIST COLORING CONJECTURE

p°?$?@ 0?B ?3 ?D¿??|ÿ?0� � � � �

Page 194: THE LIST COLORING CONJECTURE

??p ?$?@ 0?B ?4 s ?D¿??|ÿ?° ??$?@ 0? ?5 s ?ÿ¿?ÿ"?¿`?` ??P?$?D 0? ?6 s ?ÿ¿?ÿ"?¿`?®?6P?$?D � � � � ��� � � ���0? ?7 s ?ÿ¿?ÿ"?¿`?`¡??$?D 0? ?9 ?ÿ¿?ÿ"?¿`??2}� � ��� � � � ��� �

Page 195: THE LIST COLORING CONJECTURE

º?$?D 0? ?: ?ÿÿ¿?ÿ"?¿`?? }� � � ��� � �

Page 196: THE LIST COLORING CONJECTURE

?$?@ 0? ?; ??¿?ÿ"?¿`?` ?� � � ��� �   ?$?@ 0? ?< ?ÿ¿?ÿ"?¿`? ¡ ?$?@ 0? ?= ?ÿ¿?ÿ"?¿`?® 6� � � ��� � � � � � ��� �   ?$?@ 0? ? �

Page 197: THE LIST COLORING CONJECTURE

?� � ???¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 198: THE LIST COLORING CONJECTURE

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Page 199: THE LIST COLORING CONJECTURE

?� �?h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ¯???º___PPT10ª? U?`1+?D?'?¹ÿÿÿÿ=?@B? D?'?ÿÿÿÿ=?@B?A??%?(? ?B������� ��� � � � � � ��style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibility?B?style.visibilityB s ?D¿??mÿ??0 ?B ??B s ?D¿??mÿ??0 0 ?B ?@B s ?D¿??mÿ?`0 � � � � � � � � � �` ?? ??c ?B ????? ?$?Dÿÿÿ 0? ?C s ?¿?ÿ"?¿`??P?$?@ 0? ?D s ?¿?ÿ"?¿`?2?ºP?$?@ 0? ?E s ?¿?ÿ"?� � � � � � � � ��� � � � ��� � � ���¿`???c P?$?@ 0? ?F s ?¿?ÿ"?¿`? ?$?@ 0? ?G s ?¿?ÿ"?¿`?2 º ?$?@ 0? ?H s ?¿?ÿ"?¿`?? c ?$?@ 0?� � ��� � � � � ��� � � � ��� �B ?I s ?D¿??mÿ???0 ?B ?JB s ?D¿??mÿ?vv0 ?B ?K s ?D¿??mÿ? ` ?B ?L s ?D¿??mÿ??`0 ?B ?M s ?� � � � � � � � � � � � �D¿??mÿ??0 ` ?B ?N s ?D¿??mÿ? 0 ?B ?OB s ?D¿??mÿ??0 ?B ?PB s ?D¿??mÿ??0 0 ?B ?QB s ?D¿??� � � � � � � � � � � � � � �mÿ?`0 ` ?? ??? Z ?X??? Z ?$?Dÿÿÿ 0?¢� � �

Page 200: THE LIST COLORING CONJECTURE

?R ??Z¿¿?ÿ???°? ?¨{1,2}¡ 2ª� � � �� �

Page 201: THE LIST COLORING CONJECTURE

Page 202: THE LIST COLORING CONJECTURE

?S ??¿¿?ÿ?@` Z ?¨{1,2}¡ 2ª� � � �� �

Page 203: THE LIST COLORING CONJECTURE

Page 204: THE LIST COLORING CONJECTURE

?T ?(o[¿¿?ÿ?� � � ��   ` Z ?¨{1,2}¡ 2ª� �

Page 205: THE LIST COLORING CONJECTURE

Page 206: THE LIST COLORING CONJECTURE

?U ??[¿¿?ÿ?` ?Z ?¨{1,2}¡ 2ª� � � �� �

Page 207: THE LIST COLORING CONJECTURE

Page 208: THE LIST COLORING CONJECTURE

?V ?dv[¿¿?ÿ?� � � � �  ?? ?¨{1,2}¡ 2ª� �

Page 209: THE LIST COLORING CONJECTURE

Page 210: THE LIST COLORING CONJECTURE

?W ?,z[¿¿?ÿ?@? ? ?¨{1,2}¡ 2ª� � � �� �

Page 211: THE LIST COLORING CONJECTURE

?? ??? Z ?Y ???0� �

Page 212: THE LIST COLORING CONJECTURE

d ?$?Dÿÿÿ 0?¢� �

Page 213: THE LIST COLORING CONJECTURE

?Z ??[¿¿?ÿ???°? ?¨{1,2}¡ 2ª� � � �� �

Page 214: THE LIST COLORING CONJECTURE

Page 215: THE LIST COLORING CONJECTURE

?[ ?¨[¿¿?ÿ?@` Z ?¨{1,2}¡ 2ª� � � � �� �

Page 216: THE LIST COLORING CONJECTURE

Page 217: THE LIST COLORING CONJECTURE

?\ ?[¿¿?ÿ?� � ��� ��   ` Z ?¨{2,3}¡ 2ª� �

Page 218: THE LIST COLORING CONJECTURE

Page 219: THE LIST COLORING CONJECTURE

?] ?,[¿¿?ÿ?` ?Z ?¨{1,3}¡ 2ª� � � � �� �

Page 220: THE LIST COLORING CONJECTURE

Page 221: THE LIST COLORING CONJECTURE

?^ ?[¿¿?ÿ?� � �� ��  ?? ?¨{2,3}¡ 2ª� �

Page 222: THE LIST COLORING CONJECTURE

Page 223: THE LIST COLORING CONJECTURE

?_ ?[¿¿?ÿ?@? ? ?¨{1,3}¡ 2ª� � � �� �

Page 224: THE LIST COLORING CONJECTURE

? ?b ³ ?P[¿¿?ÿ?� �� � �� ��

Page 225: THE LIST COLORING CONJECTURE

??G ?$?D 0 ? � $ch(K3,3) > c?K3,3)¡ ?ÿþ ?ª&?? ?? @ ?i? ? @ ?$?Dÿÿÿ 0?¢� � �� � � �

Page 226: THE LIST COLORING CONJECTURE

?c ?0?¿¿?ÿ?? ?¨ 1¡: 2ÿþ ÿþª� � � � � � �

Page 227: THE LIST COLORING CONJECTURE

Page 228: THE LIST COLORING CONJECTURE

?d ?´[¿¿?ÿ?� � � � ��   P ?¨ 1¡: 2ÿþ ÿþª�

Page 229: THE LIST COLORING CONJECTURE

Page 230: THE LIST COLORING CONJECTURE

?e ?¤[¿¿?ÿ?@ ? ?¨ 1¡: 2ÿþ ÿþª� � � � �� �

Page 231: THE LIST COLORING CONJECTURE

Page 232: THE LIST COLORING CONJECTURE

?f ?D©[¿¿?ÿ??F @ ?¨ 2¡: 2ÿþ ÿþª� � � �� � �

Page 233: THE LIST COLORING CONJECTURE

Page 234: THE LIST COLORING CONJECTURE

?g ?<®[¿¿?ÿ?pF @ ?¨ 2¡: 2ÿþ ÿþª� � � �� �

Page 235: THE LIST COLORING CONJECTURE

Page 236: THE LIST COLORING CONJECTURE

?h ??¿¿?ÿ?0F ?@ ?¨ 2¡: 2ÿþ ÿþª� � � �� �

Page 237: THE LIST COLORING CONJECTURE

? ?�

Page 238: THE LIST COLORING CONJECTURE

?� � ?)??¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 239: THE LIST COLORING CONJECTURE

????? ? ?? ? S ?´»µ¿ÿC??¯ `}?? µ ?? ? ?¼µ¿??ÿ D?? `?$? 0?µ ?? ?�� � � ��� � � � �

Page 240: THE LIST COLORING CONJECTURE

?� � ????¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 241: THE LIST COLORING CONJECTURE

z? ??? ? ??¢

Page 242: THE LIST COLORING CONJECTURE

? s ?Xµ¿¿?ÿ??@pt ?¨Kernels and choosability¡ ( 2$ª� � � �

Page 243: THE LIST COLORING CONJECTURE

? ?

Page 244: THE LIST COLORING CONJECTURE

?� � ? ??¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � � �

Page 245: THE LIST COLORING CONJECTURE

4 ? 0? "? ? ? ?? ? S ?0?¿ÿC??¯ `}?? µ ?? ? ??¿??ÿ D?? `?$?D 0?µ ??? ? @: ? ?????$?Dÿÿÿ 0?? ?p�� � � ��� �� � � ? ??? @?2 ? s ?¿?ÿ?p ���  ??2 ? s ?¿?ÿ?p���  ??B ? ?D¿??8c?ÿ???p??¢� �

Page 246: THE LIST COLORING CONJECTURE

? ??µ¿¿?ÿ? @?: ?¨u¡ 2ª� � �� �

Page 247: THE LIST COLORING CONJECTURE

Page 248: THE LIST COLORING CONJECTURE

? ??µ¿¿?ÿ?@@: ?¨v¡ 2ª� � �� �

Page 249: THE LIST COLORING CONJECTURE

? ?" ?,°¿?ÿ "?¿`?p `� ��� ��  ?$?D 0<4º___PPT9¬ ?� � � � � The outdegree of v is d+(v)=|{u| v ? u}| The closed �neighborhood of v is N[v] = {u| v ? u or v=u } The underlying graph of D is G=(V,E ) with E = {{u,v}| u ? v or v ? u}¡:?ÿ ?P .ª? ¦?????? ?� � ��

Page 250: THE LIST COLORING CONJECTURE

?� � ?)??¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 251: THE LIST COLORING CONJECTURE

??@?$?? ? ?$? ?$ S ?´?¿ÿC??¯ `}?? µ ?? ?$ ??¿??ÿ D??? ` ?$? 0?µ ?? ?$�� � � ��� �� �

Page 252: THE LIST COLORING CONJECTURE

?� � ????¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 253: THE LIST COLORING CONJECTURE

Page 254: THE LIST COLORING CONJECTURE

?P?/ ??�

Page 255: THE LIST COLORING CONJECTURE

? ? ?? ? S ?X?¿ÿC??¯ `}?? µ ??? ?°( ¨ ?,?°( ¨ ?$?Dÿÿÿ 0? ? s ?¿?ÿ"?¿`??°X8?$?@ 0? ? s ?¿?ÿ"?¿`?? � � �� � � � � � ��� � � ���X¨ ?$?@ 0? ? s ?¿?ÿ"?¿`?� � ���  P( ??$?@ 0? ? s ?¿?ÿ"?¿`?P??$?@ 0?B ?B ?D¿??m?ÿ?`??B ?B ?D¿??m?� � ��� � � � � � � � � � �ÿ? ?B ? ?D¿??m?ÿ?`°? ?B ? ?D¿??m?ÿ?? ?B ? ?D¿??m?ÿ?0°?P ?B ?%? ?D¿??m?ÿ?` °?? ??0? ?/??� � � � � � � � � � � �� � � � � � � � �

Page 256: THE LIST COLORING CONJECTURE

0??$?Dÿÿÿ 0? ?! ?¿?ÿ"?¿`???`h?$?@ 0? ?" ?¿?ÿ"?¿`??P `??$?@ 0? ?# ?¿?ÿ"?¿`?¨0?$?@ 0? ?$ ?� � � � ��� � � � � ��� � � � � ��� � � � � � �¿?ÿ"?¿`?�� � ��

Page 257: THE LIST COLORING CONJECTURE

?$?@ 0?B ?& ?D¿??m?ÿ?`�� � � � �

Page 258: THE LIST COLORING CONJECTURE

@?°?B ?( ?D¿??m?ÿ?`� � � � �

Page 259: THE LIST COLORING CONJECTURE

? ?B ?) ?D¿??m?ÿ?`? ?B ?*? ?D¿??m?ÿ?`@?°?B ?+ ?D¿??m?ÿ?$p$ ?2 ?. ?¿?ÿ? ÿ? B`?$?Dÿÿÿ 0?� �� � � � � � � � � � � �� � � � � � � � ���� �Ԕ ?�

Page 260: THE LIST COLORING CONJECTURE

?� � ?h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ¯??j������� ��� � � �

Page 261: THE LIST COLORING CONJECTURE

?`??? ? ?? ? S ??¿ÿC??¯ `}?? ° ?? ? c ?|?0e²0e²¿¿?ÿ D?"?¿`?? `??$?D 0?° ?? ? ?p?¿¿?ÿ "?¿`?? `0� � � ��� � � �� � � � � �� ������� �� � � � � �� ��

Page 262: THE LIST COLORING CONJECTURE

?$?D 0<4º___PPT9¬ ?¨6Example: G is n-choosable if we take f(v)=n and g(v)=1¡47ÿ0?� � � � �PZ6ªN¦?????? ?� �

Page 263: THE LIST COLORING CONJECTURE

?� � ???¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � � �

Page 264: THE LIST COLORING CONJECTURE

B?p? (?? ? ?(? ?( S ?|7¾¿ÿC??¯ `}?? ¾ ?? ?( ?=¾¿?ÿ3?ÿ D??? `p ??¾ ?¦?�� � � �� �� � �  xd?² ? ( £ ?A ?¿?ÿ??¨X ???� � �� �� ?(

Page 265: THE LIST COLORING CONJECTURE

?� � ????¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 266: THE LIST COLORING CONJECTURE

??4?? ? ?4?¢� �

Page 267: THE LIST COLORING CONJECTURE

?4 ?� �聾¿¿?ÿ?°°??$?D 0 ?¨Proof: Induction on¡Hh P2i P2ª�� � �

Page 268: THE LIST COLORING CONJECTURE

?² ? 4 £ ?A ?¿?ÿ??`?x ??$?Dÿÿÿ 0?² ? 4 £ ?A ?¿?ÿ???8?�� � ��   ??$?D 0? ?4 ?¾¿¿?ÿ?� ��� ��   °?E?$?D 0 ? � b Define S:= {v?V| c?Av} Let K be a kernel of S¡N2}0 PZ2 ?ª(� �

Page 269: THE LIST COLORING CONJECTURE

? ?4 £ ?Ѝ¾¿?ÿ D??°?@ ?$? 0?¾ ?? ?4�� �� � �

Page 270: THE LIST COLORING CONJECTURE

?� � ?·??¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 271: THE LIST COLORING CONJECTURE

g?? 8?? ? ?8?¢�

Page 272: THE LIST COLORING CONJECTURE

? 8 ?� ��°?ÿ?????$?D 0 ?¡ 2 ª? ? 8 ?³¾¿¿?ÿ?¿¿¾ذ� � � � ��� �� �  ©?$?D

Page 273: THE LIST COLORING CONJECTURE

0 ?� H Define functions f ,g : V(D) ? ? ¡"} P2| �

Page 274: THE LIST COLORING CONJECTURE

P2} P2 �

Page 275: THE LIST COLORING CONJECTURE

ª2��� �

Page 276: THE LIST COLORING CONJECTURE

?² ?

Page 277: THE LIST COLORING CONJECTURE

8 £ ?A ?¿?ÿ???@ @??$?D 0?² ? 8 £ ?A ?¿?ÿ??�� � ��   ??$?D 0? ?8 ?p½¾¿¿?ÿ??°�� � �� ��  ??$?D 0 ?¨) We have that� ¡$*} P2*ª�

Page 278: THE LIST COLORING CONJECTURE

? ?8 ³ ?¼¾¿¿?ÿ?@° ?$?Dÿÿÿ 0 ?¨ i) holds, check ii)¡"i Pª2 ?² ?8 £ ?A X?¿?ÿ??x `???X$?Dÿÿÿ 0?² ?8 ?�� � �� �� � � � � � �� � �A ¿?ÿ??h���

Page 279: THE LIST COLORING CONJECTURE

h@??$?Dÿÿÿ 0? ?8� �

Page 280: THE LIST COLORING CONJECTURE

?� � ?®??¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 281: THE LIST COLORING CONJECTURE

^? ? <?? ? ?<? ? s ?4ɾ¿??ÿ D?? `??$? 0?¾ ??² ? £ ?A ?¿?ÿ???pp??$?Dÿÿÿ 0? ? < ?4�� � �� � � �� ��?$??` ?? ÿ?¿־Dÿÿÿ 0 ?� �  ¦It holds: |Bv|= g(v) and Bu'?v = ? if u,v adjacent ? G is (f:g)-choosable ? ¡<P} P2|(

Page 282: THE LIST COLORING CONJECTURE

P2} P ?���

Page 283: THE LIST COLORING CONJECTURE

? ? �� � �� �� � �

Page 284: THE LIST COLORING CONJECTURE

ª¨�

Page 285: THE LIST COLORING CONJECTURE

¦?????? ?

Page 286: THE LIST COLORING CONJECTURE

?� � ?)??¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 287: THE LIST COLORING CONJECTURE

????L?? ? ?L? ?L S ??¿ÿC???}?? ? ?? ?L ??¿??ÿ D??? `P?$?D 0?? ?? ?L� � � � � � � � � �

Page 288: THE LIST COLORING CONJECTURE

?� � ???¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � � �

Page 289: THE LIST COLORING CONJECTURE

???? ? ?? ? S ?Ї?¿ÿC??¯ `}?? ? ?? ? c ?´4¾0e²0e²¿¿?ÿ3ÿ D??° `@ ?$?Dÿÿÿ 0?? ?¦?� � � �� � � �� � � � � �� ������� �� � � � �  xd? ?� � �

Page 290: THE LIST COLORING CONJECTURE

?� � ????¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 291: THE LIST COLORING CONJECTURE

|??P?? ? ?P?¢

Page 292: THE LIST COLORING CONJECTURE

?P s ?8?¿¿?ÿ??@pt ?¨Proof of the bipartite LCC¡ ( 2$ª� � � �

Page 293: THE LIST COLORING CONJECTURE

? ?P

Page 294: THE LIST COLORING CONJECTURE

?� � ?h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ¯??S������� ��� � � �

Page 295: THE LIST COLORING CONJECTURE

? ?T?? ? ?T? ?T S ??¿ÿC??¯ `}?? ? ?? ?T ?Й?¿??ÿ D??? `` ?$?D 0?? ?? ?T ?� ��� � � � � �� � � �� ÿ ?? `??$?D?¿?؞��0<4º___PPT9¬ ?¨CExample: a kernel of S is an independent subset of S that absorbs S¡4Dÿ0?PZ <ª� � � � � � �

Page 296: THE LIST COLORING CONJECTURE

¦?????? ?T

Page 297: THE LIST COLORING CONJECTURE

?� � ????¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 298: THE LIST COLORING CONJECTURE

{?0?X?? ? ?X? ?X S ?¨?¿ÿC??¯ `}?? ? ??¢�� �

Page 299: THE LIST COLORING CONJECTURE

?X s ?4©?¿¿?ÿ? P0P ?¨Let H be a bipartite multigraph, and G:=L(H). Suppose that G is n-colorable. � � � �Then G is (kn:k)-choosable for every k. In particular G is n-choosable. ¡* 2ª� � �

Page 300: THE LIST COLORING CONJECTURE

? ?X

Page 301: THE LIST COLORING CONJECTURE

?� � ???¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � � �

Page 302: THE LIST COLORING CONJECTURE

F?@?h?? ? ?h? ?h S ??¿ÿD??P `?$? 0?? ?? ?h� � �

Page 303: THE LIST COLORING CONJECTURE

?� � ?´??¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 304: THE LIST COLORING CONJECTURE

d?P?l?? ? ?l? ?l £ ?D´?¿?ÿ D??P `�� �� �  ?$? 0?? ?? ?l�

Page 305: THE LIST COLORING CONJECTURE

?� � ?´??¯h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ������� ��� � � �

Page 306: THE LIST COLORING CONJECTURE

d?`?p?? ? ?p? ?p £ ??¿?ÿ D??P `?$? 0?? ?? ?p��� �� � � �

Page 307: THE LIST COLORING CONJECTURE

?� � ?h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ¯??d������� ��� � � �

Page 308: THE LIST COLORING CONJECTURE

?p??? ? ?? ? ?,?¿¿?ÿ?pp?$?Dÿÿÿ 0 ?� � � � �� �� � � � ¢ Corollary 1 ? D is (kn:k)-choosable "? ? G is (kn:k)-choosable for every k ?¡ªPi P2h( P2 " � � � ��� � �

Page 309: THE LIST COLORING CONJECTURE

ªb

Page 310: THE LIST COLORING CONJECTURE

)

Page 311: THE LIST COLORING CONJECTURE

? ?�

Page 312: THE LIST COLORING CONJECTURE

?� �h¿ÿ ?? h®ÿÿÿ??ÿ?ÿ¯???? 0������� ��� � �

Page 313: THE LIST COLORING CONJECTURE

???0?? ? ?0? ?0 C ?¿ÿ,?°? ?? ?? ?0 S ?,?¿ÿ,?° @� � � � �  ???

Page 314: THE LIST COLORING CONJECTURE

? ?ª? ?0�

Page 315: THE LIST COLORING CONJECTURE

?� ����h¿ÿ ?? ÿÿÿ33??ÿ²²²??? 0��� ���

Page 316: THE LIST COLORING CONJECTURE

???H?? ? ?H? ?H C ?¿ÿ,?°? ?? ?? ?H S ?,g?¿ÿ,?° @� � ��  ???

Page 317: THE LIST COLORING CONJECTURE

? ?ª? ?H�

Page 318: THE LIST COLORING CONJECTURE

?� ����h¿ÿ ?? ÿÿÿ33??ÿ²²²??? 0��� ���

Page 319: THE LIST COLORING CONJECTURE

 ??? ?? ? ? ? ?  C ?¿ÿ,?°? ?? ?? ?�   S ???¿ÿ,?° @��  ???

Page 320: THE LIST COLORING CONJECTURE

? ?� ?From the first ex we know that chi of c5 is at least 3, in the right one we find a 3-coloring ? it is 3¡$h^ ªg�

Page 321: THE LIST COLORING CONJECTURE

? ? 

Page 322: THE LIST COLORING CONJECTURE

?� ����h¿ÿ ?? ÿÿÿ33??ÿ²²²jx?MLSA?Z��� ��� � �  ?yQ1 1?(PE?J?R¼`þh¨=i$1?9J?¢?"&� � � � �  7?Y¨GT¨3?Py??fg??{ÿI?� �� ��� � ��XE§A?XKtL² '?b±hª?O????¦ÿz O4???·²´c� � � � � � � �6¡? ~ڢ ϔ?,?0&V³§\(?YJ6?¥??»?¥ÿF®C g?8£H1m??ª?0????� � � � � � ��ltC·¢m!þZ?²/yC6?S\\t??v???¯pI;?S7&?v?¯??±???? Sh ¤|?*!©? ©� � � � � � � � � �퓺 jn??A;a??|??B?5Ri|???'+U|� � � �²8)¤]?«G?_\2N;a???Qȶ£¢?[?9m@� � _e»?µ??"?P� � ꐴ=?¯ª?ڶ??c!w?_5?� � � �=U?? ??{¿? L ?g?£p·±????� ��Բ e?¿?�³?#3-?{ȠbJ?3s??,�

Page 323: THE LIST COLORING CONJECTURE

???1?£??gbXG?¸,¿??vY:dDOj~=K»"1=E²� � � � ?R}A?#�� �

Page 324: THE LIST COLORING CONJECTURE

cS¬l????6?|i�� ؏²?·??Pr::i? GB�� � ¼д?³ϑPW? ?g??!I??mN?ÿ¨\?r8Oњ3 6??$eHjA@39S?6?u? ?%i±?p?´¥?-?� �� �� �S?g¢²?½¹TZ?po???h?®Ɗ²v¿?n¥²T/ ´?}¼yY ?p?/»a;???°?sx?MlU·»N??cht1¢M"?)DjP Ӡ?ª?t?ʱ?;1¡Tʏ� � � � � � � � � � �վ ֎

Page 325: THE LIST COLORING CONJECTURE

E???)T$¤$¤ B?J?þ8¶?Qf3~??y?x}}??G?B=1µ½O¦?1½þt?v?:c?@M???b?1���� �� � � � � � � � ¾uV#V U??~¸w¿� �

Page 326: THE LIST COLORING CONJECTURE

?c?D>?\x_?.N±??`3½¼?m -9¸Sݺi»µU7?½?¾??h?4t�� �???\i ?f¢?©«?0-S±?v??9 (?2S£JJ-?o«¨ߏ{ �� � � �A�¨e²Jg???VɁK?¡?ۮ? U¦£5§?\þ|??2?µGk@%§#HC%T?*?$/ ǧ??;¦exL/:?=´K??h5¯¦A.4p~&x,m� � � � � � �� � � � � � � � �  K`¡[?�{<^^?d?k7??|?=alkS?MJ?ų4n?£j¤c??ʧZb?Kjf"<� �

Page 327: THE LIST COLORING CONJECTURE

o6~ ?¡?r{?°=E??²m?³«¸???=??¸???N+6m /[µve5?«??ǵC3?#?CK¥B:±XU:G?&$O,SG=� � � � � � ��� �£??;?®x; (Eݢ£c??^¿?8??¼>� �

Page 328: THE LIST COLORING CONJECTURE

²??r?8??L>¡2l=¶ $مi§)ª?A?xNµx?MlE~³»þ?? !qM??F?N?!ªSʡ� � � � � �� �?T .U"¨ T~.=)-·.?©@²P?v?j· h\¨¦??ٸ� � ��� � � � � � �·??7????Qÿȳ ?? 2f@@0?¹\???J?$\? ?°k»#ÿ_9xM?iҥ¡?cE?\?%^??¥·� � � � � � � �ւ #«ya¨w?Ly¿+?1 ?i¿Z#� � �

Page 329: THE LIST COLORING CONJECTURE

þA´<)?? b ?)?µ?c?^i??p&p?E³??'_&} ?oµÿ@s� � � �

Page 330: THE LIST COLORING CONJECTURE

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Page 331: THE LIST COLORING CONJECTURE

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Page 333: THE LIST COLORING CONJECTURE

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Page 334: THE LIST COLORING CONJECTURE

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Page 337: THE LIST COLORING CONJECTURE

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Page 338: THE LIST COLORING CONJECTURE

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Page 342: THE LIST COLORING CONJECTURE

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Page 348: THE LIST COLORING CONJECTURE

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Page 355: THE LIST COLORING CONJECTURE

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