Application of the Representation Theory of Symmetric Groups for the Computation of Chromatic Polynomials of Graphs Mikhail Klin Christian Pech 1 Department of Mathematics Ben Gurion University of the Negev Beer Sheva, Israel 2008-01-01 1 supported by the Skirball postodctoral fellowship of the Center of Advanced Studies in Mathematics at the Mathematics Department of Ben Gurion University. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 1 / 27
Outline 1 2 3 4 Representations of V(Sym(x),Ω x ) Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 2 / 27
Brief History of the Project Oberwolfach, 1994: a general idea suggested to Klin by P. de la Harpe and R. Bacher 1994 1995: first outline by Klin 1997 1998: use of representations of association schemes a la Zieschang by Klin-Pech Grenoble, 1998, Workshop in memory of F. Jaeger: discussion of a new version with de la Harpe and Bacher 2000 2001: Use of representations of symmetric groups by Pech July 2001: Exchange of ideas between N. Biggs and Klin Fall 2001: Joint paper in memory of J.J. Seidel submitted by Biggs, Klin and P. Reinfeld Fall 2007: Preparation of the final version Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 3 / 27
Graphs Definition A graph Γ is a pair (V, E) where V is a finite set, E ( V 2) = {{v, w} v, w V, v w}, V = n is the order of Γ, we assume V = n = {0, 1,..., n 1}. Example 0 4 9 5 6 1 8 7 3 2 Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 4 / 27
Vertex colorings Given a graph Γ = (V, E). Definition Coloring of Γ with s colors: φ : V s = {0, 1,..., s 1}. proper coloring: v, w V : {v, w} E φ(v) φ(w). Example 0 5 4 9 6 1 8 7 3 2 Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 5 / 27
Color-partitions of Γ = (V, E) V V equivalence relation such that (v, w) {v, w} / E if φ is a proper coloring of Γ with s colors then φ = ker φ = {(v, w) V 2 φ(v) = φ(w)} is a color-partition of Γ with r s classes. for each color-partition with r classes there are s(s 1) (s r + 1) proper colorings φ of Γ with s r colors and with ker φ =. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 6 / 27
Chromatic polynomial of Γ = (V, E) where Clear: C(Γ; x) = s m r u (r) (x) r=1 m r : # of color-partitions of Γ with r classes, u (r) (x) = x(x 1) (x r + 1) C(Γ; s) is equal to the number of proper colorings of Γ with s colors. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 7 / 27
Deletion & Contraction Method Γ = (V, E), e = {v, w} E Γ (e) deletion of e: e v Γ (e) contraction of e: w v w v e w v=w C(Γ; x) = C(Γ (e) ; x) C(Γ (e) ; x) Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 8 / 27
Computing C(Γ; x) by deletion and contraction Advantages: general and simple, easily implemented Disadvantage: in worst case 2 E steps are required in order to find C(Γ; x). Complexity: Deciding whether C(Γ, x) = 0 is NP-complete for all natural numbers x > 2. Hence in general, computing the chromatic polynomial is hard. Polynomial algorithms for special classes of graphs are possible (and wanted). Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 9 / 27
Generalized ladder graphs Given a basic graph Γ 0 = (V 0, E 0 ). Definition Γ = (V, E) is called generalized ladder for Γ 0 if V = V 0 V 1 V n 1, i : Γ Vi = Γ0, {v, w} E i : {v, w} V i (v V i w V i+1(mod n) ) We identify V with n V 0 and assume i n : {(i, v),(i, w)} E {v, w} E 0 Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 10 / 27
(0,0) (5,0) (0,2) (0,1) (1,0) (5,2) (5,1) (1,2) (1,1) (4,0) (2,0) (4,2) (4,1) (3,0) (2,2) (2,1) (3,2) (3,1) Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 11 / 27
Coloring generalized ladders φ be a proper coloring of Γ 0. Then φ i : (i, v) φ(v) is a proper coloring of Γ Vi. Ω x be the set of all proper colorings of Γ 0 with x colors. A proper coloring of Γ corresponds to a tuple (φ 0,...,φ n 1 ) Ω n x Any (φ 0,...,φ n 1 ) Ω n x induces a coloring of Γ: φ : (i, v) φ i (v) However, φ might be improper! Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 12 / 27
Coloring generalized ladders φ be a proper coloring of Γ 0. Then φ i : (i, v) φ(v) is a proper coloring of Γ Vi. Ω x be the set of all proper colorings of Γ 0 with x colors. A proper coloring of Γ corresponds to a tuple (φ 0,...,φ n 1 ) Ω n x Any (φ 0,...,φ n 1 ) Ω n x induces a coloring of Γ: φ : (i, v) φ i (v) However, φ might be improper! Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 12 / 27
Coloring generalized ladders φ be a proper coloring of Γ 0. Then φ i : (i, v) φ(v) is a proper coloring of Γ Vi. Ω x be the set of all proper colorings of Γ 0 with x colors. A proper coloring of Γ corresponds to a tuple (φ 0,...,φ n 1 ) Ω n x Any (φ 0,...,φ n 1 ) Ω n x induces a coloring of Γ: φ : (i, v) φ i (v) However, φ might be improper! Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 12 / 27
Coloring generalized ladders φ be a proper coloring of Γ 0. Then φ i : (i, v) φ(v) is a proper coloring of Γ Vi. Ω x be the set of all proper colorings of Γ 0 with x colors. A proper coloring of Γ corresponds to a tuple (φ 0,...,φ n 1 ) Ω n x Any (φ 0,...,φ n 1 ) Ω n x induces a coloring of Γ: φ : (i, v) φ i (v) However, φ might be improper! Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 12 / 27
Coloring generalized ladders φ be a proper coloring of Γ 0. Then φ i : (i, v) φ(v) is a proper coloring of Γ Vi. Ω x be the set of all proper colorings of Γ 0 with x colors. A proper coloring of Γ corresponds to a tuple (φ 0,...,φ n 1 ) Ω n x Any (φ 0,...,φ n 1 ) Ω n x induces a coloring of Γ: φ : (i, v) φ i (v) However, φ might be improper! Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 12 / 27
Question: Which elements of Ω n x correspond to proper colorings of Γ? For i n define R (i) x Ω 2 x : R (i) x = {(φ,ψ) v V i, w V i+1(mod n) : Proposition {v, w} E φ(v) ψ(w)} (φ 0,...,φ n 1 ) corresponds to a proper coloring of Γ iff i n : (φ i,φ i+1(mod n) ) R (i) x The R (i) x are called compatibility-relations. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 13 / 27
Theorem Let B (i) x = A(R (i) x ) be the adjacency matrix of R (i) x (i n). Then C(Γ; x) = trace(b (0) x B (1) x B (n 1) x ) Main tool for the proof: Lemma Let Ω = {0, 1,...,n 1} and let R, S Ω 2. Let A = A(R), B = A(S), C = AB = (c i,j ) then c i,j = {k (i, k) R, (k, j) S} Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 14 / 27
Problem: How to find trace(b (0) x B (1) x B (n 1) x ) efficiently (without multiplying huge matrices)? Idea: take some semisimple matrix-algebra A x over C that contains all B (i) x (i n), use ordinary representation theory in order to find the trace. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 15 / 27
Analysis of the colorings of the basic graph Γ 0 = (V 0, E 0 ) Assume V 0 = m = {0, 1,..., m 1}. Ω x the set of all proper vertex colorings of Γ 0 with x colors. Each coloring φ x V 0 = x m. Ω x is an m-ary relation of x = {0, 1,...,x 1}, the R (i) x are binary relations over Ω x. Proposition Ω x is an invariant relation of Sym(x) the symmetric group. Proposition Let (φ,ψ) R (i) x. Then π Sym(x) : (φ π,ψ π ) R (i) x. That is R (i) x is a binary invariant relation of (Sym(x),Ω x ). Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 16 / 27
An algebra for the B (i) x Remark R 1,...,R r be the minimal binary invariant relations (2-orbits in the sense of H. Wielandt) of (Sym(x),Ω x ), their adjacency matrices A 1,..., A r form the standard basis of a coherent algebra (in the sense of D.G. Higman), It is denoted: clearly B (i) x = A(R (i) x ) A x V(Sym(x),Ω x ) =: A x V(Sym(x),Ω x ) consists of all matrices that commute with all permutation matrices of (Sym(x),Ω x ). It is called centralizer algebra of (Sym(x),Ω x ). Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 17 / 27
Interesting properties of A x = V(Sym(x), Ω x ) General Properties A x is semisimple, the degrees and the number of the irreducible representations are independent of x, the multiplicities are polynomials in x, the matrices of the irreducible representations can be given as polynomial matrices. The irreducible representations of A x are closely related to those of (Sym(x),Ω x ) The bases of the representation modules may be described by embedding Specht-generators into C(Ω x ) Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 18 / 27
Interesting properties of A x = V(Sym(x), Ω x ) General Properties A x is semisimple, the degrees and the number of the irreducible representations are independent of x, the multiplicities are polynomials in x, the matrices of the irreducible representations can be given as polynomial matrices. The irreducible representations of A x are closely related to those of (Sym(x),Ω x ) The bases of the representation modules may be described by embedding Specht-generators into C(Ω x ) Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 18 / 27
Consequences for computing chromatic polynomials we saw that C(Γ; x) = trace(b x (0) B x (1) B x (n 1) ) for certain matrices B (i) x A x = V(Sym(x),Ω x ) Let C(Ω x ) = U 1 U 2 U m be a decomposition into irreducible A x -modules. Let σ j (B (i) x ) be the corresponding representation matrix of B (i) x on U j. Theorem C(Γ; x) = m trace(σ j (B (0) x )σ j (B (1) x ) σ j (B (n 1) x )). j=1 Polynomial Complexity Computing C(Γ; x) is reduced to the computation of traces of certain products of polynomial matrices whose order is independent of x. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 19 / 27
(0,0) (5,0) (0,2) (0,1) (1,0) (5,2) (5,1) (1,2) (1,1) (4,0) (2,0) (4,2) (4,1) (3,0) (2,2) (2,1) (3,2) (3,1) Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 20 / 27
Proper colorings of the basis graph Ω x = {(a, b, c) {a, b, c} = 3, {a, b, c} {0, 1,..., x 1}} The compatibility-relations R (i) x R (0) x = {((a 0, a 1, a 2 ),(b 0, b 1, b 2 )) a 1 b 0 }, R x (1) = {((a 0, a 1, a 2 ),(b 0, b 1, b 2 )) a 1 b 1 and a 2 b 2 }, R (2) x = {((a 0, a 1, a 2 ),(b 0, b 1, b 2 )) a i b i (i = 0, 1, 2)}, R (3) x = R (0) x, R x (4) = R x (1), R (5) x = R (2) x. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 21 / 27
The irreducible representations of B (0) x, B (1) x,and B (2) x i σi(b (0) x ) σi(b (1) x ) σi(b (2) x ) µi 0 (x 1) 2 (x 2) (x 2 3x + 3)(x 2) x 3 6x 2 + 14x 13 1 1 0 0 0 (x 2)(x 1) x 2 x 2 0 0 0 0 0 0 x 3 (x 2) 2 (x 2) x 3 x 2 (x 2) 2 x 2 + 5x 7 x 3 x 3 x 3 x 2 + 5x 7 x 3 x-1 x 3 x 3 x 2 + 5x 7 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 x-2 x 3 1 1 1 x 3 1 1 2x(x 3) 1 1 x 3 3 0 0 1 x 2 6x + 7 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 x-2 x 3 1 1 1 x 3 1 1 2 (x 2)(x 1) 1 1 x 3 5 ( 0 ) 0 ( 0 ) 0 ( 1 ) 0 0 0 0 0 0 1 1 3 (x 4)(x 2)x 1 6 0 0 1 6 (x 3)(x 2)(x 1) Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 22 / 27
(0,0) (5,0) (0,2) (0,1) (1,0) (5,2) (5,1) (1,2) (1,1) (4,0) (2,0) (4,2) (4,1) (3,0) (2,2) (2,1) (3,2) (3,1) C(Γ, x) = (x 14 25x 13 + 296x 12 2.200x 11 + 11.468x 10 44.364x 9 + 131.430x 8 303.281x 7 + 548.914x 6 778.115x 5 + 855.040x 4 712.148x 3 + 430.029x 2 171.041x + 34.320)(x 2) 2 (x 1)x. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 23 / 27
Decomposition of C(Ω x ) Decomposition into constituents C(Sym(x),Ω x ) and V(Sym(x),Ω x ) share the primitive (mutually orthogonal) idempotents. The images of the primitive idempotents on C(Ω x ) are the constituents. To each partition of x there corresponds a constituent. Each partition is given by a Young-diagram. Irreducible representations of (Sym(x),Ω x ) Each constituent splits (uniquely) into an orthogonal sum of mutually isomorphic irreducible submodules. Each such irreducible submodule is isomorphic to the Specht-module defined by the corresponding partition. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 24 / 27
Irreducible representations of V(Sym(x), Ω x ) Given a fixed partition π of x (e.g. in terms of a Young-diagram), S the Specht module for this partition, V = S S S the corresponding constituent, e i : S V the corresponding embeddings (i = 1,...,m). S is generated by the Specht-generators: S = s 1, s 2,..., s d. where d equals to the number of standard Young-tableaus for π Proposition Define U i := e 1 (s i ), e 2 (s i ),..., e m (s i ). Then 1 the U i are mutually isomorphic irreducible representation-modules of V(Sym(x),Ω x ), and 2 V = U 1 U 2 U d is a complete decomposition of V into irreducible submodules. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 25 / 27
We defined the class of generalized ladder graphs. We counted proper colorings of generalized ladder graphs as trace of a product of certain compatibility-matrices. We identified the compatibility-matrices as elements of the centralizer algebra of the symmetric group acting on proper colorings of the base graph. We demonstrated how using the representation theory of the symmetric groups the compatibility matrices can be block-diagonalized where the order of each block is independent of the number of colors. From this we derived an efficient method for computing chromatic polynomials of generalized ladder graphs. Our approach inspired further developments by Biggs. There is potential to extend out method to graphs of bounded tree-width. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 26 / 27
References N. L. BIGGS, R. M. DAMERELL, AND D. A. SANDS, Recursive families of graphs. J. Combinatorial Theory Ser. B 12, (1972), 123 131. N. L. BIGGS, Algebraic Graph Theory. 2nd Edition, Cambridge Univesity Press, 1994. N. L. BIGGS, M. H. KLIN, AND P. REINFELD, Algebraic methods for chromatic polynomials. Research Report LSE-CDAM-2001-06, Centre for Discrete and Applicable Mathematics, Sep. 2001. N. BIGGS, Chromatic polynomials and representations of the symmetric group. Linear Algebra and its Applications 356, (2002), 3 26. N. BIGGS, Specht modules and chromatic polynomials. J. Combinatorial Theory (B) 92, (2004), 359 377. N. BIGGS, M. KLIN, AND P. REINFELD, Algebraic methods for chromatic polynomials. Europ. J. Combinatorics 25, (2004), 147 160. Klin,Pech Chromatic polynomials via Representation Theory 2008-01-01 27 / 27