Combinatorial Commutative Algebra, Graph Colorability, and Algorithms

Transcription

1 I G,k = g 1,...,g n Combinatorial Commutative Algebra, Graph Colorability, and Algorithms Chris Hillar (University of California, Berkeley) Joint with Troels Windfeldt! (University of Copenhagen)!

2 Outline of talk Introduction: Colorability Motivational Problem Previous Work: k-colorability algebraic characterizations Groebner bases Notation and Color Encodings Statement of Main Results Algorithms

3 " Graph Colorings G Let G be a simple graph G = (V,E ) with vertices V = {1,,...,n}, edges E Def: A k-coloring of G is an assignment of k colors to the vertices of G Def: A k-coloring is proper if adjacent vertices receive different colors Def: A graph is k-colorable if it has a proper k-coloring

4 Unique Colorability A -colorable graph that is not -colorable: Other than permuting the colors, no other proper -colorings Def: A uniquely k-colorable graph is one which has exactly k! proper k-colorings. In this case, the coloring is unique up to permutation of colors 4

5 Technical Observation G cannot be uniquely k-colorable if there is a proper k-coloring not using all k colors Not uniquely -colorable although there are! = 6 proper -colorings of G that use all colors 5

6 Motivational Problem In 1990, Xu showed that uniquely k-colorable graphs must have many edges Theorem (Xu): If G is uniquely k-colorable then E > (k-1)n - k(k-1)/ In example, 7 > 5 - = 7 Notice: G contains a triangle 6

7 Xu s Conjecture In general, he conjectured that if equality holds, G must contain a k-clique. Conjecture (Xu): If G is uniquely k-colorable and E = (k-1)n - k(k-1)/, then G contains a k-clique This conjecture was shown to be false in 001 by Akbari et al using a combinatorial argument We wanted to find a JPE proof (Just Press Enter) 7

8 Computer Proof This leads to the following concrete problem: Problem: Find an (effective) algorithm to decide unique k-colorability (and find the coloring). -colorable? Uniquely -colorable? 8

9 Computer Proof This leads to the following concrete problem: Problem: Find an (effective) algorithm to decide unique k-colorability (and find the coloring). -colorable? Uniquely -colorable? 9

10 Computer Proof This leads to the following concrete problem: Problem: Find an (effective) algorithm to decide unique k-colorability (and find the coloring). -colorable? Uniquely -colorable? Y e s! 10

11 Computer Proof This leads to the following concrete problem: Problem: Find an (effective) algorithm to decide unique k-colorability (and find the coloring). -colorable? Uniquely -colorable? Y e s! " " Y e s! 11

12 Computer Proof This leads to the following concrete problem: Problem: Find an (effective) algorithm to decide unique k-colorability (and find the coloring). -colorable? Uniquely -colorable? Y e s! " " Y e s!... 1

13 Counterexample to Xu How about the graph actually used by Akbari, Mirrokni, Sadjad: E = 45, n = 4, k = 1

14 Algebraic Colorability Main Idea (implicit in work of Bayer, de Loera, Lovász): colorings are points in varieties varieties are represented by ideals ideals can be manipulated with Groebner Bases 14

15 k-colorings as Points in Varieties Setup: F is an algebraically closed field, (char F ) k So F contains k distinct kth roots of unity. Let I k = x 1 k - 1, x k - 1,..., x n k - 1 F [x 1,...,x n ] This ideal is radical, and V(I k ) = k n Can think of point v = (v 1,...,v n ) in V(I k ) as assignment v = (v 1,...,v n ) vertex i gets color v i Eg. If 1 = Blue, -1 = Red, then v = (1,-1,1) is coloring 1 15

16 Proper k-colorings of Graphs We can also restrict to proper k-colorings of graph G I G,k = I k + x k-1 i + x k- i x j + + x k-1 j : (i,j ) E This ideal is radical, and V(I G,k ) = # proper k-colorings Proof: (<=) If v proper, then v i v j and (v i - v j ) (v i k v j k-1 ) = v i k - v j k = 1-1 = 0 Thus, v i k v j k-1 = 0 and v in V(I G,k ) 16

17 Algebraic Characterization Notice that if I G,k = 1 = F [x 1,...,x n ] then V(I G,k ) = => G is not k-colorable Therefore, we have a test for k-colorability: Algorithm: Compute a reduced Groebner basis B for I G,k. Then, B = {1} iff G is not k-colorable. 1 " " " " I G,k = x 1-1, x - 1, x - 1, " " " " " x 1 + x, x + x, x 1 + x (k = ) " " = 1 x 1 = (x 1 -x )(x 1 +x ) + (x -x )(x +x ) + (x 1 -x )(x 1 +x ) 17

18 Graph Polynomial The graph polynomial of G encodes adjacency of vertices algebraically: " " "f G = {i,j } E i <j (x i x j ) One should think of f G as a test polynomial for k-colorability. " " I = x 1-1, x - 1, x - 1 k =, n = " " " " 1 Notice that f G = (x 1 -x )(x -x )(x 1 -x ) I k Conclusion: There is a -coloring that is proper! 18

19 Characterization Theorem k-colorability Theorem (Bayer, de Loera, Alon, Tarsi, Mnuk, Kleitman, Lovász): The following are equivalent "(1) G is not k-colorable "() I G,k = 1 "() f G is contained in I k (colorings zero f G ) Corollary: There are simple tests for k-colorability involving polynomial algebra. Our goal: Develop a similar characterization theorem for unique k-colorability and give a complete description of I G,k when G is k-colorable 19

20 Preparation: Color classes Given G which is k-colorable with a coloring v = (v 1,...,v n ) using all k colors, we define: Color class of i = cl(i ) = { j : v j = v i } representative of cl(i ) = max{ j : j is in cl(i ) } Denote these representatives 1 " "m 1 < m <... < m k = n Eg. cl(1) = {1,5}, cl() = {,} 4 " cl(4) = {4} m 1 = < m = 4 < m = 5 0 5

21 New Polynomial Encoding We need a replacement for the graph polynomial f G in the statement of the k-colorability theorem Def: Let U {1,...,n }. Then we set h U d to be the sum of all monomials of degree d in the variables {x i : i in U } Eg. U = {1,,}, d = h U d = x 1 + x + x + x 1 x + x 1 x + x x Also, we set h U 0 = 1 1

22 Replacements for f G Def: Given a proper k-coloring v, for each vertex i, let " " " x k i - 1 " " i = m k (= n) " "g i = " h j {mj,...,mk} " "i = m j for some j k " " " h 1 {i,m...,mk} "i in cl(m 1 ) " " " x i - x max cl(i) "otherwise g 5 = x 5-1 g 4 = h {4,5} = x 4 + x 4 x 5 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g 1 " = x 1 - x 5 (m 1,m,m )= (,4,5) 1 4 5

23 Can go backwards Find a proper k-coloring giving a set " " " x k i - 1 " " i = m k (= n) " "g i = " h j {mj,...,mk} " "i = m j for some j k " " " h 1 {i,m...,mk} "i in cl(m 1 ) " " " x i - x max cl(i) "otherwise g 5 = x 5-1 g 4 = h {4,5} = x 4 + x 4 x 5 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g 1 " = x 1 - x 5 (m 1,m,m )= (?,?,?) 1 4 5

24 Can go backwards Find a proper k-coloring giving a set " " " x k i - 1 " " i = m k (= n) " "g i = " h j {mj,...,mk} " "i = m j for some j k " " " h 1 {i,m...,mk} "i in cl(m 1 ) " " " x i - x max cl(i) "otherwise g 5 = x 5-1 g 4 = h {4,5} = x 4 + x 4 x 5 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g 1 " = x 1 - x 5 (m 1,m,m )= (?,?,5)

25 Can go backwards Find a proper k-coloring giving a set " " " x k i - 1 " " i = m k (= n) " "g i = " h j {mj,...,mk} " "i = m j for some j k " " " h 1 {i,m...,mk} "i in cl(m 1 ) " " " x i - x max cl(i) "otherwise g 5 = x 5-1 g 4 = h {4,5} = x 4 + x 4 x 5 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g 1 " = x 1 - x 5 (m 1,m,m )= (?,4,5)

26 Can go backwards Find a proper k-coloring giving a set " " " x k i - 1 " " i = m k (= n) " "g i = " h j {mj,...,mk} " "i = m j for some j k " " " h 1 {i,m...,mk} "i in cl(m 1 ) " " " x i - x max cl(i) "otherwise g 5 = x 5-1 g 4 = h {4,5} = x 4 + x 4 x 5 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g 1 " = x 1 - x 5 (m 1,m,m )= (,4,5)

27 Can go backwards Find a proper k-coloring giving a set " " " x k i - 1 " " i = m k (= n) " "g i = " h j {mj,...,mk} " "i = m j for some j k " " " h 1 {i,m...,mk} "i in cl(m 1 ) " " " x i - x max cl(i) "otherwise g 5 = x 5-1 g 4 = h {4,5} = x 4 + x 4 x 5 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g 1 " = x 1 - x 5 (m 1,m,m )= (,4,5)

28 Can go backwards Find a proper k-coloring giving a set " " " x k i - 1 " " i = m k (= n) " "g i = " h j {mj,...,mk} " "i = m j for some j k " " " h 1 {i,m...,mk} "i in cl(m 1 ) " " " x i - x max cl(i) "otherwise g 5 = x 5-1 g 4 = h {4,5} = x 4 + x 4 x 5 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g = h 1 {,4,5} = x + x 4 + x 5 g 1 " = x 1 - x 5 (m 1,m,m )= (,4,5)

29 Encoding: v --> A v = <g 1,...,g n > The set of polynomials {g 1,...,g n } encodes the coloring v Lemma: Let A v = g 1,...,g n "(1) I G,k A v "() A v is a radical ideal "() V(A v ) = k! Interpretation: "(1) zeroes of A v are proper k-colorings of G "() A v and its zeroes are in 1-1 correspondence "() Up to permutation, A v encodes precisely v 9

30 Characterization Theorem Theorem [-,W 06]: Let G be a graph. The following are equivalent "(1) G is k-colorable "() A v I G,k "() A v = I G,k Point: We have found an interpretation involving ideals for the statement: " { V(A v ) : v is proper } = { proper colorings } 0

31 Unique Characterization In general, the map from proper k-colorings v --> A v = <g 1,...,g n > only depends on how v partitions V = {1,...,n} into color classes. In particular, Fact: If G is uniquely colorable, then there is a unique set of polynomials {g 1,...,g n } that corresponds to all v. Proof: All v partition V the same way 1

32 Unique Characterization Corollary [-,W 06]: Fix a proper k-coloring v of G. Let A v = g 1,...,g n. The following are equivalent "(1) G is uniquely k-colorable "() g 1,...,g n belong to I G,k " () A v = g 1,...,g n = I G,k More canonically, we have the following Theorem [-,W 06]: G is uniquely k-colorable if and only if the reduced Groebner basis for I G,k (w.r.t any term order with x n < < x 1 ) has the form g 1,...,g n

33 Algorithms The main theorems give algorithms for determining unique colorability Algorithm 1: Given a proper k-coloring, construct the polynomials g 1,...,g n and reduce them modulo I G,k Algorithm : Compute the reduced Groebner basis for I G,k and see whether it has the form g 1,...,g n ; if so, read off the coloring. - (easy to check form and to read off coloring)

34 Just Press Enter Computing with a field F with char F =, we find that (after pressing enter and waiting 5 seconds) the following graph is indeed uniquely k-colorable. 4

35 The End (of talk) 5