Coloring and List Coloring of Graphs and Hypergraphs

Transcription

1 Coloring and List Coloring of Graphs and Hypergraphs Douglas B. West Department of Mathematics University of Illinois at Urbana-Champaign

2 Graphs Def. graph G: a vertex set V(G) and an edge set E(G), where each edge is an unordered pair of vertices.

3 Graphs Def. graph G: a vertex set V(G) and an edge set E(G), where each edge is an unordered pair of vertices. Ex. bipartite graph: V(G) is two independent sets.

4 Graphs Def. graph G: a vertex set V(G) and an edge set E(G), where each edge is an unordered pair of vertices. Ex. bipartite graph: V(G) is two independent sets. Ex. planar graph: drawable without edge crossings.

5 Graphs Def. graph G: a vertex set V(G) and an edge set E(G), where each edge is an unordered pair of vertices. Ex. bipartite graph: V(G) is two independent sets. Ex. What do the colors mean? planar graph: drawable without edge crossings.

6 Coloring Def. coloring of G: assigns each vertex a label (color). proper coloring c: E(G) c() = c(). G is k-colorable if proper coloring using k colors. chromatic number χ(g) = min{k : G is k-colorable}.

7 Coloring Def. coloring of G: assigns each vertex a label (color). proper coloring c: E(G) c() = c(). G is k-colorable if proper coloring using k colors. chromatic number χ(g) = min{k : G is k-colorable}. Ex. How many time slots are needed to schedule Senate committees? common members different time slots

8 Coloring Def. coloring of G: assigns each vertex a label (color). proper coloring c: E(G) c() = c(). G is k-colorable if proper coloring using k colors. chromatic number χ(g) = min{k : G is k-colorable}. Ex. How many time slots are needed to schedule Senate committees? common members different time slots Let V(G) = {committees} E(G) = {pairs of vertices w. common members}. min #time slots = χ(g).

9 Coloring Def. coloring of G: assigns each vertex a label (color). proper coloring c: E(G) c() = c(). G is k-colorable if proper coloring using k colors. chromatic number χ(g) = min{k : G is k-colorable}. Ex. How many time slots are needed to schedule Senate committees? common members different time slots Let V(G) = {committees} E(G) = {pairs of vertices w. common members}. min #time slots = χ(g). Here χ(g) = 4

10 List Coloring Vizing [1976], Erdős Rubin Taylor [1979] Some committees can t meet at certain times.

11 List Coloring Vizing [1976], Erdős Rubin Taylor [1979] Some committees can t meet at certain times. Def. list assignment: L() = color set available at. L-coloring: proper coloring s.t. c() L() V(G). k-choosable G: L-coloring whenever all L() k. list chromatic number (or choosability) χ l (G) = min{k : G is k-choosable}.

12 List Coloring Vizing [1976], Erdős Rubin Taylor [1979] Some committees can t meet at certain times. Def. list assignment: L() = color set available at. L-coloring: proper coloring s.t. c() L() V(G). k-choosable G: L-coloring whenever all L() k. list chromatic number (or choosability) χ l (G) = min{k : G is k-choosable}. Prop. χ(g) χ l (G) Δ(G) + 1. Pf. Lower bound: The lists may be identical. Upper bound: Lists of size Δ(G) + 1 a color is available for each successive vertex in any order. d() = degree of vertex ; Δ(G) = mx V(G) d().

13 χ l (G) May Exceed χ(g) Ex. χ(k 2,4 ) = 2, but χ l (K 2,4 ) > 2. {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} K r,s = complete bipartite graph, part-sizes r and s.

14 χ l (G) May Exceed χ(g) Ex. χ(k 2,4 ) = 2, but χ l (K 2,4 ) > 2. {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} Bipartite graphs may have large choice number. Prop. If m = 2k 1 k, then Km,m is not k-choosable.

15 χ l (G) May Exceed χ(g) Ex. χ(k 2,4 ) = 2, but χ l (K 2,4 ) > 2. {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} Bipartite graphs may have large choice number. Prop. If m = 2k 1 k, then Km,m is not k-choosable. Pf. Use the k-sets in [2k 1] as the lists for both X and Y. X 1 m Y y 1 y m

16 χ l (G) May Exceed χ(g) Ex. χ(k 2,4 ) = 2, but χ l (K 2,4 ) > 2. {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} Bipartite graphs may have large choice number. Prop. If m = 2k 1 k, then Km,m is not k-choosable. Pf. Use the k-sets in [2k 1] as the lists for both X and Y. X 1 m Y y 1 y m < k colors on X some vertex of X left uncolored. k colors on X vertex of Y w. that list is uncolorable.

17 Planar Graphs Vizing [1976] Conj. ( E R T [1979] ) mx{χ l(g): G is planar} = 5. Ex. Non 4-choosable: Voigt [1993] 238 vertices Mirzakhani [1996] 63 vertices

18 Planar Graphs Vizing [1976] Conj. ( E R T [1979] ) mx{χ l(g): G is planar} = 5. Ex. Non 4-choosable: Voigt [1993] 238 vertices Mirzakhani [1996] 63 vertices Thm. (Thomassen [1994]) χ l (G) 5 if G is planar.

19 Planar Graphs Vizing [1976] Conj. ( E R T [1979] ) mx{χ l(g): G is planar} = 5. Ex. Non 4-choosable: Voigt [1993] 238 vertices Mirzakhani [1996] 63 vertices Thm. (Thomassen [1994]) χ l (G) 5 if G is planar. Pf. Idea: Prove stronger result (by induction). Coloring still choosable if L() = 3 for outer vertices, except L() = 1 for two adjacent outer vertices.

20 Planar Graphs Vizing [1976] Conj. ( E R T [1979] ) mx{χ l(g): G is planar} = 5. Ex. Non 4-choosable: Voigt [1993] 238 vertices Mirzakhani [1996] 63 vertices Thm. (Thomassen [1994]) χ l (G) 5 if G is planar. Pf. Idea: Prove stronger result (by induction). Coloring still choosable if L() = 3 for outer vertices, except L() = 1 for two adjacent outer vertices. We may assume that all bounded faces are triangles and the outer face is a cycle.

21 Planar Graphs Vizing [1976] Conj. ( E R T [1979] ) mx{χ l(g): G is planar} = 5. Ex. Non 4-choosable: Voigt [1993] 238 vertices Mirzakhani [1996] 63 vertices Thm. (Thomassen [1994]) χ l (G) 5 if G is planar. Pf. Idea: Prove stronger result (by induction). Coloring still choosable if L() = 3 for outer vertices, except L() = 1 for two adjacent outer vertices. We may assume that all bounded faces are triangles and the outer face is a cycle. b Basis step: bc

22 Induction Step Let [ 1,..., p ] be the outer cycle C in order, with L( 1 ) = L( p ) = 1.

23 Induction Step Let [ 1,..., p ] be the outer cycle C in order, with L( 1 ) = L( p ) = 1. Case 1: C has a chord. 1 Case 2: C has no chord p G 1 j G 2 p

24 Induction Step Let [ 1,..., p ] be the outer cycle C in order, with L( 1 ) = L( p ) = 1. Case 1: C has a chord. 1 Case 2: C has no chord p G 1 j G 2 p Case 1: Choose L-coloring on G 1, then L -coloring on G 2 using the colors chosen from L for and j.

25 Induction Step Let [ 1,..., p ] be the outer cycle C in order, with L( 1 ) = L( p ) = 1. Case 1: C has a chord. 1 p G 1 j G 2 p Case 2: C has no chord G 2 Case 1: Choose L-coloring on G 1, then L -coloring on G 2 using the colors chosen from L for and j. m Case 2: L( 1 ) = {}. Pick, y L( 2 ) {}. Delete and y from each L( ). Choose L-coloring on G 2. Extend to 2 using or y.

26 Alon Tarsi: A tool for upper bounds on χ l (G) Def. circulation C in a digraph D = a subdigraph s.t. indegree d C () = outdegree d+ () at each V(D). C

27 Alon Tarsi: A tool for upper bounds on χ l (G) Def. circulation C in a digraph D = a subdigraph s.t. indegree d C () = outdegree d+ () at each V(D). C diff(d) = #circulations of even size in D #circulations of odd size in D

28 Alon Tarsi: A tool for upper bounds on χ l (G) Def. circulation C in a digraph D = a subdigraph s.t. indegree d C () = outdegree d+ () at each V(D). C diff(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 diff(d) = 0, then G has an L-coloring whenever L() > d + () for all V(G). D

29 Alon Tarsi: A tool for upper bounds on χ l (G) Def. circulation C in a digraph D = a subdigraph s.t. indegree d C () = outdegree d+ () at each V(D). C diff(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 diff(d) = 0, then G has an L-coloring whenever L() > d + () for all V(G). D Ex. even cycle 2 even, 0 odd χ l (C 2k ) 2

30 Alon Tarsi: A tool for upper bounds on χ l (G) Def. circulation C in a digraph D = a subdigraph s.t. indegree d C () = outdegree d+ () at each V(D). C diff(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 diff(d) = 0, then G has an L-coloring whenever L() > d + () for all V(G). D Ex. even cycle Ex. odd cycle 2 even, 0 odd 1 even, 1 odd 1 even, 0 odd χ l (C 2k ) 2 no info χ l (C 2k+1 ) 3

31 Hall s Theorem Def. matching = a set of pairwise disjoint edges. X, Y-bigraph = bipartite graph with partite sets X and Y. X Y S

32 Hall s Theorem Def. matching = a set of pairwise disjoint edges. X, Y-bigraph = bipartite graph with partite sets X and Y. X Y S Thm. (P. Hall [1935]) TONCAS

33 Hall s Theorem Def. matching = a set of pairwise disjoint edges. X, Y-bigraph = bipartite graph with partite sets X and Y. X Y S Thm. (P. Hall [1935]) TONCAS An X, Y-bigraph G has a matching covering X N(S) S for all S X.

34 Hall s Theorem Def. matching = a set of pairwise disjoint edges. X, Y-bigraph = bipartite graph with partite sets X and Y. X Y S Thm. (P. Hall [1935]) TONCAS An X, Y-bigraph G has a matching covering X N(S) S for all S X. Pf. Nec.: Observed above. Suff.: Induction on X.

35 Hall s Theorem Def. matching = a set of pairwise disjoint edges. X, Y-bigraph = bipartite graph with partite sets X and Y. X Y S Thm. (P. Hall [1935]) TONCAS An X, Y-bigraph G has a matching covering X N(S) S for all S X. Pf. Nec.: Observed above. Suff.: Induction on X. Case 1: N(S) > S for all S with S X. Match one vertex arbitrarily, delete that pair, apply induction hypothesis.

36 Hall s Theorem Def. matching = a set of pairwise disjoint edges. X, Y-bigraph = bipartite graph with partite sets X and Y. X Y S Thm. (P. Hall [1935]) TONCAS An X, Y-bigraph G has a matching covering X N(S) S for all S X. Pf. Nec.: Observed above. Suff.: Induction on X. Case 1: N(S) > S for all S with S X. Match one vertex arbitrarily, delete that pair, apply induction hypothesis. Case 2: N(S) = S for some S with S X.

37 Case 2 for Hall s Theorem Case 2: N(S) = S for some S with S X. X S Y N(S)

38 Case 2 for Hall s Theorem Case 2: N(S) = S for some S with S X. Let G = subgraph induced by S N(S). S S N G (S ) = N G (S ) S, so G satisfies H.C. X G S Y N(S)

39 Case 2 for Hall s Theorem Case 2: N(S) = S for some S with S X. Let G = subgraph induced by S N(S). S S N G (S ) = N G (S ) S, so G satisfies H.C. X G H S T Y N(S) N H (T) Let H = subgraph induced by (X S) (Y N(S)). To prove H satisfies H.C., compute N H (T) = N(T S) N(S) T S S = T.

40 Case 2 for Hall s Theorem Case 2: N(S) = S for some S with S X. Let G = subgraph induced by S N(S). S S N G (S ) = N G (S ) S, so G satisfies H.C. X G H S T Y N(S) N (T) Let H = subgraph induced by (X S) (Y N(S)). To prove H satisfies H.C., compute N H (T) = N(T S) N(S) T S S = T. Use matchings from G and H (induction hypothesis).

41 Balanced Orientations of Graphs Lem. (Tarsi) Every graph G has an orientation D such that Δ + E(H) (D) mx H G. V(H)

42 Balanced Orientations of Graphs Lem. (Tarsi) Every graph G has an orientation D such that Δ + E(H) (D) mx H G. V(H) E(H) Pf. Let ρ = mx H G. Form X, Y-bigraph from G. V(H) X = E(G) j 1 1 j ρ ρ j Y = V(G) [ρ]

43 Balanced Orientations of Graphs Lem. (Tarsi) Every graph G has an orientation D such that Δ + E(H) (D) mx H G. V(H) E(H) Pf. Let ρ = mx H G. Form X, Y-bigraph from G. V(H) X = E(G) j 1 1 j k ρ ρ j Y = V(G) [ρ] Orient j in D when j matched to k. matching covering X orientn. D with Δ + (D) ρ.

44 Balanced Orientations of Graphs Lem. (Tarsi) Every graph G has an orientation D such that Δ + E(H) (D) mx H G. V(H) E(H) Pf. Let ρ = mx H G. Form X, Y-bigraph from G. V(H) X = E(G) j 1 1 j k ρ ρ j Y = V(G) [ρ] Orient j in D when j matched to k. matching covering X orientn. D with Δ + (D) ρ. For S X, H G with S = E(H) and N(S) = V(H) [ρ]. Compute N(S) = V(H) ρ E(H) = S.

45 Balanced Orientations of Graphs Lem. (Tarsi) Every graph G has an orientation D such that Δ + E(H) (D) mx H G. V(H) E(H) Pf. Let ρ = mx H G. Form X, Y-bigraph from G. V(H) X = E(G) j 1 1 j k ρ ρ j Y = V(G) [ρ] Orient j in D when j matched to k. matching covering X orientn. D with Δ + (D) ρ. For S X, H G with S = E(H) and N(S) = V(H) [ρ]. Compute N(S) = V(H) ρ E(H) = S. Hall s Theorem matching covering X.

46 Choosability of Planar Bipartite Graphs Thm. (Alon Tarsi [1992]) Every bipartite planar graph is 3-choosable.

47 Choosability of Planar Bipartite Graphs Thm. (Alon Tarsi [1992]) Every bipartite planar graph is 3-choosable. Pf. Euler s Formula: For a connected plane graph, #vertices n #edges m + #faces f = 2.

48 Choosability of Planar Bipartite Graphs Thm. (Alon Tarsi [1992]) Every bipartite planar graph is 3-choosable. Pf. Euler s Formula: For a connected plane graph, #vertices n #edges m + #faces f = 2. Bipartite planar H Every face has length at least 4, so 2m 4f, so Euler s Formula m 2n 4.

49 Choosability of Planar Bipartite Graphs Thm. (Alon Tarsi [1992]) Every bipartite planar graph is 3-choosable. Pf. Euler s Formula: For a connected plane graph, #vertices n #edges m + #faces f = 2. Bipartite planar H Every face has length at least 4, so 2m 4f, so Euler s Formula m 2n 4. ρ 2, and orientation D with Δ + (D) 2.

50 Choosability of Planar Bipartite Graphs Thm. (Alon Tarsi [1992]) Every bipartite planar graph is 3-choosable. Pf. Euler s Formula: For a connected plane graph, #vertices n #edges m + #faces f = 2. Bipartite planar H Every face has length at least 4, so 2m 4f, so Euler s Formula m 2n 4. ρ 2, and orientation D with Δ + (D) 2. Every circulation D in D decomposes into cycles. Every cycle in G has even length E(D ) is even.

51 Choosability of Planar Bipartite Graphs Thm. (Alon Tarsi [1992]) Every bipartite planar graph is 3-choosable. Pf. Euler s Formula: For a connected plane graph, #vertices n #edges m + #faces f = 2. Bipartite planar H Every face has length at least 4, so 2m 4f, so Euler s Formula m 2n 4. ρ 2, and orientation D with Δ + (D) 2. Every circulation D in D decomposes into cycles. Every cycle in G has even length E(D ) is even. diff(d) = 0. Alon Tarsi Theorem G is 3-choosable.

52 Hypergraphs Def. hypergraph: a vertex set V(H) and edge set E(H), with each edge being any subset of V(H). A hypergraph is k-uniform if every edge has size k.

53 Hypergraphs Def. hypergraph: a vertex set V(H) and edge set E(H), with each edge being any subset of V(H). A hypergraph is k-uniform if every edge has size k. Def. For hypergraphs, the definitions of coloring proper coloring k-colorable chromatic number list assignment L-coloring k-choosable list chromatic number are exactly the same as for the special case of graphs, except that we must rephrase one: A proper coloring is a coloring with no monochromatic edge.

54 The Probabilistic Method Idea: The Existence Argument. Build an object by a random experiment in which a desired property corresponds to some event A. If Prob(A) > 0, then in some outcome A occurs, so some object has the desired property.

55 The Probabilistic Method Idea: The Existence Argument. Build an object by a random experiment in which a desired property corresponds to some event A. If Prob(A) > 0, then in some outcome A occurs, so some object has the desired property. Thm. Every k-uniform hypergraph with fewer than 2 k 1 edges is 2-colorable. Pf. Color randomly, giving color 0 or 1 to each vertex with probability 1/ 2 each, independently.

56 The Probabilistic Method Idea: The Existence Argument. Build an object by a random experiment in which a desired property corresponds to some event A. If Prob(A) > 0, then in some outcome A occurs, so some object has the desired property. Thm. Every k-uniform hypergraph with fewer than 2 k 1 edges is 2-colorable. Pf. Color randomly, giving color 0 or 1 to each vertex with probability 1/ 2 each, independently. Prob(a given edge is monochromatic) = 1/2 k 1. Prob(some edge is monochromatic) #edges/2 k 1 < 1. coloring with no monochromatic edge.

57 Application to Bipartite Choosability Cor. If G is an n-vertex X, Y-bigraph, then χ l (G) 1 + lg n.

58 Application to Bipartite Choosability Cor. If G is an n-vertex X, Y-bigraph, then χ l (G) 1 + lg n. Pf. Show that G is k-choosable when k > 1 + lg n.

59 Application to Bipartite Choosability Cor. If G is an n-vertex X, Y-bigraph, then χ l (G) 1 + lg n. Pf. Show that G is k-choosable when k > 1 + lg n. Given list assignment L on G with each L() = k, form hypergraph H with V(H) = V(G) L(). Let E(H) have one edge for each vertex in G: its list!

60 Application to Bipartite Choosability Cor. If G is an n-vertex X, Y-bigraph, then χ l (G) 1 + lg n. Pf. Show that G is k-choosable when k > 1 + lg n. Given list assignment L on G with each L() = k, form hypergraph H with V(H) = V(G) L(). Let E(H) have one edge for each vertex in G: its list! If n < 2 k 1 (that is, k > 1 + lg n), then H is 2-colorable. Call the two colors X and Y. This restricts colors in V(H) to usage on X or Y in G.

61 Application to Bipartite Choosability Cor. If G is an n-vertex X, Y-bigraph, then χ l (G) 1 + lg n. Pf. Show that G is k-choosable when k > 1 + lg n. Given list assignment L on G with each L() = k, form hypergraph H with V(H) = V(G) L(). Let E(H) have one edge for each vertex in G: its list! If n < 2 k 1 (that is, k > 1 + lg n), then H is 2-colorable. Call the two colors X and Y. This restricts colors in V(H) to usage on X or Y in G. For each V(G), we must choose a color from L(). When X, choose a color restricted to X. When Y, choose a color restricted to Y.

62 Face Hypergraphs of Planar Graphs Def. face hypergraph of a plane graph G = hypergr. H with V(H) = V(G) and E(H) ={vertex sets of faces of G} y z E(H) = {y, yz, yz}

63 Face Hypergraphs of Planar Graphs Def. face hypergraph of a plane graph G = hypergr. H with V(H) = V(G) and E(H) ={vertex sets of faces of G} y z E(H) = {y, yz, yz} Thm. (Ramamurthi [2001]) The face hypergraph H of any plane graph G is 3-choosable.

64 Face Hypergraphs of Planar Graphs Def. face hypergraph of a plane graph G = hypergr. H with V(H) = V(G) and E(H) ={vertex sets of faces of G} y z E(H) = {y, yz, yz} Thm. (Ramamurthi [2001]) The face hypergraph H of any plane graph G is 3-choosable. Pf. May assume all faces of G are triangles.

65 Face Hypergraphs of Planar Graphs Def. face hypergraph of a plane graph G = hypergr. H with V(H) = V(G) and E(H) ={vertex sets of faces of G} y z E(H) = {y, yz, yz} Thm. (Ramamurthi [2001]) The face hypergraph H of any plane graph G is 3-choosable. Pf. May assume all faces of G are triangles. Given 3-lists at vertices, choose a proper coloring of H.

66 Face Hypergraphs, continued Need to choose colors from lists s.t. 2 on each face.

67 Face Hypergraphs, continued Need to choose colors from lists s.t. 2 on each face. Add A and B to each list to make lists of size 5. 5-choosable G has proper coloring from these lists.

68 Face Hypergraphs, continued Need to choose colors from lists s.t. 2 on each face. Add A and B to each list to make lists of size 5. 5-choosable G has proper coloring from these lists. Faces not having both A and B are okay. c A d

69 Face Hypergraphs, continued Need to choose colors from lists s.t. 2 on each face. Add A and B to each list to make lists of size 5. 5-choosable G has proper coloring from these lists. Faces not having both A and B are okay. c A d A B Vertices with A or B chosen form bipartite subgraph G, since proper coloring was chosen for G.

70 Face Hypergraphs, continued Need to choose colors from lists s.t. 2 on each face. Add A and B to each list to make lists of size 5. 5-choosable G has proper coloring from these lists. Faces not having both A and B are okay. c A d A B Vertices with A or B chosen form bipartite subgraph G, since proper coloring was chosen for G. proper coloring of G chosen from the original 3-lists on these vertices, since it is a bipartite planar graph.

71 Face Hypergraphs, continued Need to choose colors from lists s.t. 2 on each face. Add A and B to each list to make lists of size 5. 5-choosable G has proper coloring from these lists. Faces not having both A and B are okay. c A d A B Vertices with A or B chosen form bipartite subgraph G, since proper coloring was chosen for G. proper coloring of G chosen from the original 3-lists on these vertices, since it is a bipartite planar graph. Conj. (Ramamurthi) Face hypergraphs are 2-choosable.

72 Algebraic Interpretation of List Coloring Idea: Express proper coloring in terms of a polynomial.

73 Algebraic Interpretation of List Coloring Idea: Express proper coloring in terms of a polynomial. Def. For a graph G with vertices numbered 1,..., n, the graph polynomial f G () is { j E(G): <j} ( j ). Numbers s 1,..., s n chosen for the vertices form a proper coloring if and only if f G () = 0.

74 Algebraic Interpretation of List Coloring Idea: Express proper coloring in terms of a polynomial. Def. For a graph G with vertices numbered 1,..., n, the graph polynomial f G () is { j E(G): <j} ( j ). Numbers s 1,..., s n chosen for the vertices form a proper coloring if and only if f G () = 0. Consider list assignment L for G with L( ) = S. L-coloring exists f G (s) = 0 for some s S 1 S n.

75 Algebraic Interpretation of List Coloring Idea: Express proper coloring in terms of a polynomial. Def. For a graph G with vertices numbered 1,..., n, the graph polynomial f G () is { j E(G): <j} ( j ). Numbers s 1,..., s n chosen for the vertices form a proper coloring if and only if f G () = 0. Consider list assignment L for G with L( ) = S. L-coloring exists f G (s) = 0 for some s S 1 S n. Recall: polynomial of degree d has at most d zeros.

76 Algebraic Interpretation of List Coloring Idea: Express proper coloring in terms of a polynomial. Def. For a graph G with vertices numbered 1,..., n, the graph polynomial f G () is { j E(G): <j} ( j ). Numbers s 1,..., s n chosen for the vertices form a proper coloring if and only if f G () = 0. Consider list assignment L for G with L( ) = S. L-coloring exists f G (s) = 0 for some s S 1 S n. Recall: polynomial of degree d has at most d zeros. Lem. (Combinatorial Nullstellensatz; Alon [1999]) Let f be homogen. polynomial of degree m in n varbs. If the coefficient of n =1 t is nonzero and each S > t, then s n =1 S such that f (s) = 0.

77 Graph Polynomial and Orientations f G ( 1,..., n ) = ( j ) { j E(G): <j}

78 Graph Polynomial and Orientations f G ( 1,..., n ) = { j E(G): <j} ( j ) = c d Term in expansion choose or j from each factor choose tail for each edge, forming D d d

79 Graph Polynomial and Orientations f G ( 1,..., n ) = { j E(G): <j} ( j ) = c d Term in expansion choose or j from each factor choose tail for each edge, forming D Exponent on is d + D ( ). Sign of orientn. = parity of #edges picking larger index. d d

80 Graph Polynomial and Orientations f G ( 1,..., n ) = { j E(G): <j} ( j ) = c d Term in expansion choose or j from each factor choose tail for each edge, forming D Exponent on is d + D ( ). Sign of orientn. = parity of #edges picking larger index. coeff.( d ) = d d #even orientn w outdeg. d 1,..., d n #odd orientn w outdeg. d 1,..., d n

81 Graph Polynomial and Orientations f G ( 1,..., n ) = { j E(G): <j} ( j ) = c d Term in expansion choose or j from each factor choose tail for each edge, forming D d d Exponent on is d + D ( ). Sign of orientn. = parity of #edges picking larger index. coeff( d ) = #even orientn w outdeg. d 1,..., d n #odd orientn w outdeg. d 1,..., d n = #even-sized circulations in D #odd-sized circulations in D, where D is a fixed orientation w outdegrees d 1,..., d n.

82 Graph Polynomial and Orientations f G ( 1,..., n ) = { j E(G): <j} ( j ) = c d Term in expansion choose or j from each factor choose tail for each edge, forming D d d Exponent on is d + D ( ). Sign of orientn. = parity of #edges picking larger index. coeff( d ) = #even orientn w outdeg. d 1,..., d n #odd orientn w outdeg. d 1,..., d n = #even-sized circulations in D #odd-sized circulations in D, where D is a fixed orientation w outdegrees d 1,..., d n. Bijection needed!

83 Concluding the Alon-Tarsi Theorem Bijection: Given a fixed orientation D with outdegrees d 1,..., d n, each orientation D with the same outdegrees as D corresponds to a circulation in D whose edges are those reversed in D.

84 Concluding the Alon-Tarsi Theorem Bijection: Given a fixed orientation D with outdegrees d 1,..., d n, each orientation D with the same outdegrees as D corresponds to a circulation in D whose edges are those reversed in D. D D edges of D reversed by D

85 Concluding the Alon-Tarsi Theorem Bijection: Given a fixed orientation D with outdegrees d 1,..., d n, each orientation D with the same outdegrees as D corresponds to a circulation in D whose edges are those reversed in D. D D edges of D reversed by D diff(d) = 0 coeff( d ) = 0 f (s) = 0 for some s S where each S is fixed with S > d χ l (G) 1 + Δ + (D)

86 Alon Tarsi for Hypergraphs Plan of action (for a k-uniform hypergraph H): 0. Order the vertices 1,..., n. 1. Define a hypergraph polynomial f H such that f H () = 0 precisely when coloring with for all gives the same number to all vertices of some edge. 2. Generalize graph orientation to hypergraphs to interpret exponents in monomials. 3. Nullstellensatz If lists are bigger than the corresponding exponents in a monomial term in f H with nonzero coefficient, then H has an L-coloring. 4. Interpret coefficients in terms of some structure we can count to obtain a sufficient condition for an orientation of H to guarantee L-coloring.

87 The Hypergraph Polynomial (k prime) Def. For a k-uniform hypergraph H with vertices 1,..., n, the hypergraph polynomial f H is defined by f H ( 1,..., n ) = ( 0 + θ θ k 1 k 1 ), 0 k 1 E(H) where θ is a kth root of unity and the vertices of each edge are written in increasing order of indices.

88 The Hypergraph Polynomial (k prime) Def. For a k-uniform hypergraph H with vertices 1,..., n, the hypergraph polynomial f H is defined by f H ( 1,..., n ) = ( 0 + θ θ k 1 k 1 ), 0 k 1 E(H) where θ is a kth root of unity and the vertices of each edge are written in increasing order of indices. When numbers 1,..., n are assigned to 1,..., n, f H ( 1,..., n ) = 0 numbers are equal on some edge.

89 The Hypergraph Polynomial (k prime) Def. For a k-uniform hypergraph H with vertices 1,..., n, the hypergraph polynomial f H is defined by f H ( 1,..., n ) = ( 0 + θ θ k 1 k 1 ), 0 k 1 E(H) where θ is a kth root of unity and the vertices of each edge are written in increasing order of indices. When numbers 1,..., n are assigned to 1,..., n, f H ( 1,..., n ) = 0 numbers are equal on some edge. Def. An orientation of a hypergraph chooses one source vertex in each edge. Terms like θ j d in the expansion of f H () come from orientations with chosen as the source in d edges.

90 The Hypergraph List Coloring Theorem Thm. (Ramamurthi West [2005]) For prime k, let D be an orientation of a k-uniform hypergraph H such that the number of balanced partitions with modular sum j is not the same for all j. If each L( ) is larger than the number of edges in which D chooses as the source, then H has an L-coloring.

91 The Hypergraph List Coloring Theorem Thm. (Ramamurthi West [2005]) For prime k, let D be an orientation of a k-uniform hypergraph H such that the number of balanced partitions with modular sum j is not the same for all j. If each L( ) is larger than the number of edges in which D chooses as the source, then H has an L-coloring. Pf. Follow the Plan!