the list coloring conjecture
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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 PresentationTRANSCRIPT
Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Theory of Combinatorial Algorithms Prof. Emo Welzl and Prof. Bernd Gärtner
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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 �
? 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?� �� ��� �
¨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�� � �� � �� �
?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� � �
? ¨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? �����
?F#¨example¡ªª?� �
¨list coloring conjecture ¡ª&� � Pch(G) = c?G) whenever G is a line graph ¡L)ªt��
?ªª?¨ definitions¡� � �
ª � Consider a digraph D=(V,E) Notation: u ? v means that (u,v) ? E:¡¬B ªN � ����
?¨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¡$ÿþ *ª?� � ��� �� �� �
?H%¨example¡ªª?G$¨(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��� � �� � �� ��� ���� �� �� � ��� �����
?¨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�� �
?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¬?�� �� � ��� ��� �
?&ªª?*ª� � � � Induction hypothesis ? G is (f :g )-choosable This means: $?v ?Av\{c} with |Bv |= g (v), s.t. Bv '?v = ? if u,v adjacent Define ¡
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?:¨(m:n)-choosable¡ª* ¨bG is called (m:n)-choosable if it is (f:g)-choosable for the constant functions � �f(v)=m and g(v)=n ¡<cZ
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?J'¨ corollary 1¡�
ª* ¨¦Let D be a digraph in which the maximum outdegree is n-1 every induced subdigraph has a kernel �Then D is (kn:k)-choosable for every k. In particular D is n-choosable.¡jIdC??dH??dBIªB �
?<ªª?=¨ definitions¡� � �
ª � An orientation of a graph G is any digraph having G as underying graph Let K be a set of vertices in a �digraph. Then K absorbs a vertex v if N[v]'? ¹??. K absorbs a set S if K absorbs every vertex of S¡<?Z ÿþ, �� ª? ?>¨the bipartite LCC¡ª* ª??ª���� �� ����� � � � � � ¤Proof: Let V := V(G) = E(H) Define Ax:={v?V| v is incident with x, x?V(H)}. Ax is called row if x?X, column if x?Y If v is a vertex of G, then R(v) is the row and C(v) the column containing v Take a proper coloring f: V ? {1,...,n} of V Define an orientation D of G in which u ? v if R(u)=R(v) and f(u)>f(v) or C(u)=C(v) and f(u)<f(v)¡^·0 " Z2¶0 " Z25·0 " Z2? ? 0%4ª )���� �� � ����� �
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?@ª� � ?Check i) and ii) of Corollary 1 i): ? (since f is one-to-one on N[v]) ii): induction on |S| Given S?V, show the existence of a kernel in S Define T := {v?S| f(v)<f(u) whenever v¹? ? R(v)'?} T absorbs S If T is independent, then it s a kernel ?¡|¶ " 2-· " 26¶ " 2$ =0 6 ªp� �� � �� ������ ����� � �� �
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?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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