Coloring and List Coloring of Graphs and Hypergraphs Douglas B. West Department of Mathematics University of Illinois at Urbana-Champaign [email protected] Graphs Def. graph G: a vertex set +(G) and an edge set E(G), where each edge is an unordered pair of vertices. Graphs Def. graph G: a vertex set +(G) and an edge set E(G), where each edge is an unordered pair of vertices. Ex. bipartite graph: +(G) is two independent sets. • • • • • • • • • • • • Graphs Def. graph G: a vertex set +(G) and an edge set E(G), where each edge is an unordered pair of vertices. Ex. bipartite graph: +(G) is two independent sets. • • • • • • • • • • • • Ex. planar graph: drawable without edge crossings. • • • • • • • • • • Graphs Def. graph G: a vertex set +(G) and an edge set E(G), where each edge is an unordered pair of vertices. Ex. bipartite graph: +(G) is two independent sets. • • • • • • • • • • • • Ex. planar graph: drawable without edge crossings. • • • What do the colors • • • • mean? • • • Coloring Def. coloring of G: assigns each vertex a label (color). proper coloring 4: 2 E(G) ) 4() 6= 4(). G is <-colorable if ! proper coloring using ≤ < colors. chromatic number |(G) = GCH{< : G is <-colorable}. Coloring Def. coloring of G: assigns each vertex a label (color). proper coloring 4: 2 E(G) ) 4() 6= 4(). G is <-colorable if ! proper coloring using ≤ < colors. chromatic number |(G) = GCH{< : G is <-colorable}. Ex. How many time slots are needed to schedule Senate committees? • common members ) different time slots Coloring Def. coloring of G: assigns each vertex a label (color). proper coloring 4: 2 E(G) ) 4() 6= 4(). G is <-colorable if ! proper coloring using ≤ < colors. chromatic number |(G) = GCH{< : G is <-colorable}. Ex. How many time slots are needed to schedule Senate committees? • common members ) different time slots Let +(G) = {committees} E(G) = {pairs of vertices w. common members}. GCH #time slots = |(G). Coloring Def. coloring of G: assigns each vertex a label (color). proper coloring 4: 2 E(G) ) 4() 6= 4(). G is <-colorable if ! proper coloring using ≤ < colors. chromatic number |(G) = GCH{< : G is <-colorable}. Ex. How many time slots are needed to schedule Senate committees? • common members ) different time slots Let +(G) = {committees} E(G) = {pairs of vertices w. common members}. GCH #time slots = |(G). • • • Here |(G) = 4 • • • • • • • List Coloring Vizing [1976], Erdos–R˝ ubin–Taylor [1979] Some committees can't meet at certain times. List Coloring Vizing [1976], Erdos–R˝ ubin–Taylor [1979] Some committees can't meet at certain times. Def. list assignment: !() = color set available at . !-coloring: proper coloring s.t. 4() 2 !() 2 +(G). <-choosable G: ! !-coloring whenever all j!()j ≥ <. list chromatic number (or choosability) |ℓ(G) = GCH{< : G is <-choosable}. List Coloring Vizing [1976], Erdos–R˝ ubin–Taylor [1979] Some committees can't meet at certain times. Def. list assignment: !() = color set available at . !-coloring: proper coloring s.t. 4() 2 !() 2 +(G). <-choosable G: ! !-coloring whenever all j!()j ≥ <. list chromatic number (or choosability) |ℓ(G) = GCH{< : G is <-choosable}. Prop. |(G) ≤ |ℓ(G) ≤ m(G) + 1. Pf. Lower bound: The lists may be identical. Upper bound: Lists of size m(G) + 1 ) a color is available for each successive vertex in any order. • 5() = degree of vertex ; m(G) = G}R2+(G) 5(). |ℓ(G) May Exceed |(G) Ex. |( 2,4) = 2, but |ℓ( 2,4) > 2. •{1, 2} {1, 3}• {1, 4}• •{2, 3} •{2, 4} •{3, 4} • C,D = complete bipartite graph, part-sizes C and D. |ℓ(G) May Exceed |(G) Ex. |( 2,4) = 2, but |ℓ( 2,4) > 2. •{1, 2} {1, 3}• {1, 4}• •{2, 3} •{2, 4} •{3, 4} Bipartite graphs may have large choice number. Prop. If > = 2<−1 , then is not <-choosable. < >,> |ℓ(G) May Exceed |(G) Ex. |( 2,4) = 2, but |ℓ( 2,4) > 2. •{1, 2} {1, 3}• {1, 4}• •{2, 3} •{2, 4} •{3, 4} Bipartite graphs may have large choice number. Prop. If > = 2<−1 , then is not <-choosable. < >,> Pf. Use the <-sets in [2<−1] as the lists for both - and .. 1 > - • • • • • • . • • • • • • J1 J> |ℓ(G) May Exceed |(G) Ex. |( 2,4) = 2, but |ℓ( 2,4) > 2. •{1, 2} {1, 3}• {1, 4}• •{2, 3} •{2, 4} •{3, 4} Bipartite graphs may have large choice number. Prop. If > = 2<−1 , then is not <-choosable. < >,> Pf. Use the <-sets in [2<−1] as the lists for both - and .. 1 > - • • • • • • . • • • • • • J1 J> < < colors on - ) some vertex of - left uncolored. < colors on - ) vertex of . w. that list is uncolorable. Planar Graphs Vizing [1976] Conj. ( E–R–T [1979] ) G}R{|ℓ(G): G is planar} = 5. Voigt [1993] 238 vertices Ex. Non 4-choosable: Mirzakhani [1996] 63 vertices Planar Graphs Vizing [1976] Conj. ( E–R–T [1979] ) G}R{|ℓ(G): G is planar} = 5. Voigt [1993] 238 vertices Ex. Non 4-choosable: Mirzakhani [1996] 63 vertices Thm. (Thomassen [1994]) |ℓ(G) ≤ 5 if G is planar. Planar Graphs Vizing [1976] Conj. ( E–R–T [1979] ) G}R{|ℓ(G): G is planar} = 5. Voigt [1993] 238 vertices Ex. Non 4-choosable: Mirzakhani [1996] 63 vertices Thm. (Thomassen [1994]) |ℓ(G) ≤ 5 if G is planar. Pf. Idea: Prove stronger result (by induction). Coloring still choosable if j!()j = 3 for outer vertices, except j!()j = 1 for two adjacent outer vertices. Planar Graphs Vizing [1976] Conj. ( E–R–T [1979] ) G}R{|ℓ(G): G is planar} = 5. Voigt [1993] 238 vertices Ex. Non 4-choosable: Mirzakhani [1996] 63 vertices Thm. (Thomassen [1994]) |ℓ(G) ≤ 5 if G is planar. Pf. Idea: Prove stronger result (by induction). Coloring still choosable if j!()j = 3 for outer vertices, except j!()j = 1 for two adjacent outer vertices. We may assume that all bounded faces are triangles and the outer face is a cycle. • • • • • • • • • • ! • ! • • • • • • • • • • • Planar Graphs Vizing [1976] Conj. ( E–R–T [1979] ) G}R{|ℓ(G): G is planar} = 5. Voigt [1993] 238 vertices Ex. Non 4-choosable: Mirzakhani [1996] 63 vertices Thm. (Thomassen [1994]) |ℓ(G) ≤ 5 if G is planar. Pf. Idea: Prove stronger result (by induction). Coloring still choosable if j!()j = 3 for outer vertices, except j!()j = 1 for two adjacent outer vertices. We may assume that all bounded faces are triangles and the outer face is a cycle. •3 Basis step: • • 34 Induction Step Let [1, . , A] be the outer cycle C in order, with j!(1)j = j!(A)j = 1. Induction Step Let [1, . , A] be the outer cycle C in order, with j!(1)j = j!(A)j = 1. Case 1: C has a chord. Case 2: C has no chord. 2 • 1• • 1• •3 A• G1 G2 A• • ; Induction Step Let [1, . , A] be the outer cycle C in order, with j!(1)j = j!(A)j = 1. Case 1: C has a chord. Case 2: C has no chord. 2 • 1• • 1• •3 A• G1 G2 A• • ; 0 Case 1: Choose !-coloring on G1, then ! -coloring on G2 using the colors chosen from ! for and ;. Induction Step Let [1, . , A] be the outer cycle C in order, with j!(1)j = j!(A)j = 1. Case 1: C has a chord. Case 2: C has no chord. 2 • 1• • 1• •3 • • • A• G1 G2 A• 1 > • G − 2 ; 0 Case 1: Choose !-coloring on G1, then ! -coloring on G2 using the colors chosen from ! for and ;. Case 2: !(1) = {}. Pick , J 2 !(2) − {}. Delete and J from each !(). Choose !-coloring on G − 2. Extend to 2 using or J. Alon–Tarsi: A tool for upper bounds on |ℓ(G) Def. circulation C in a digraph D = a subdigraph s.t. indegree 5−() = outdegree 5+() at each 2 +(D). C C Alon–Tarsi: A tool for upper bounds on |ℓ(G) Def. circulation C in a digraph D = a subdigraph s.t. indegree 5−() = outdegree 5+() at each 2 +(D). C C >C@@(D) = #circulations of even size in D − #circulations of odd size in D Alon–Tarsi: A tool for upper bounds on |ℓ(G) Def. circulation C in a digraph D = a subdigraph s.t. indegree 5−() = outdegree 5+() at each 2 +(D). C C >C@@(D) = #circulations of even size in D − #circulations of odd size in D Thm. (Alon–Tarsi [1992]) If G has an orientation D such that >C@@(D) 6= 0, then G has an !-coloring whenever j!()j > 5+() for all 2 +(G). D Alon–Tarsi: A tool for upper bounds on |ℓ(G) Def. circulation C in a digraph D = a subdigraph s.t. indegree 5−() = outdegree 5+() at each 2 +(D). C C >C@@(D) = #circulations of even size in D − #circulations of odd size in D Thm. (Alon–Tarsi [1992]) If G has an orientation D such that >C@@(D) 6= 0, then G has an !-coloring whenever j!()j > 5+() for all 2 +(G). D Ex. even cycle • • • • • • 2 even, 0 odd |ℓ(C2< ) ≤ 2 Alon–Tarsi: A tool for upper bounds on |ℓ(G) Def. circulation C in a digraph D = a subdigraph s.t. indegree 5−() = outdegree 5+() at each 2 +(D). C C >C@@(D) = #circulations of even size in D − #circulations of odd size in D Thm. (Alon–Tarsi [1992]) If G has an orientation D such that >C@@(D) 6= 0, then G has an !-coloring whenever j!()j > 5+() for all 2 +(G). D Ex. even cycle Ex. odd cycle • • • • • • • • • • • • • • • • 2 even, 0 odd 1 even, 1 odd 1 even, 0 odd |ℓ(C2< ) ≤ 2 no info |ℓ(C2<+1) ≤ 3 Hall's Theorem Def. matching = a set of pairwise disjoint edges.