The Tutte polynomial is a generalization of the chromatic polynomial of graph
colorings. Here we present an extension called the rooted Tutte polynomial,
which is defined on a graph where one or more vertices are colored with
prescribed colors. We establish a number of results pertaining to the rooted
Tutte polynomial, including a duality relation in the case that all roots
reside around a single face of a planar graph. The connection with the Potts
model is also reviewed.Comment: plain latex, 14 pages, 2 figs., to appear in Annales de l'Institut
  Fourier (1999

King, C.

Lu, W. T.

Wu, F. Y.

English

arXiv

Numérisation de Documents Anciens Mathématiques

On the rooted Tutte polynomial

F. Y. Wu

C. King

W. T. Lu

Crossref

Annales de l’institut Fourier

Annales de l’institut Fourier (AIF)

ANNALES DE L’INSTITUT FOURIERF. Y. WUC. KINGW. T. LUOn the rooted Tutte polynomialAnnales de l’institut Fourier, tome 49, no 3 (1999), p. 1103-1114<http://www.numdam.org/item?id=AIF_1999__49_3_1103_0>© Annales de l’institut Fourier, 1999, tous droits réservés.L’accès aux archives de la revue « Annales de l’institut Fourier »(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-tion commerciale ou impression systématique est constitutive d’une in-fraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Ann. Inst. Fourier, Grenoble49, 3 (1999), 1103-1114ON THE ROOTED TUTTE POLYNOMIAL (*)by F.Y. WU, C. KING and W.T. LU1. The Tutte polynomial.Consider a finite graph G with vertex set V and edge set E. A spanningsubgraph G\S) C G' is a subgraph of G containing all members of V andan edge set S C E. Let C be a set of q distinct colors. A q- coloring of G isa coloring of the vertices in V such that two vertices connected by an edgebear different colors. It is well-known that the number of g-colorings of Gis given by the chromatic polynomial (see [1])(1) P(G-,q)=^q^\-l^,SCEwhere p(S) is the number of components in the spanning subgraph G^S).Alternately, we can regard (1) as generating colorings of components ofspanning subgraphs of G with q colors with an edge weight —1.As an extension of the chromatic polynomial, Tutte [2], [3], [4] intro-duced what is now known as the Tutte polynomial(2) Q{G'^v)= ̂  ̂ )^|-M+p(^^t,v)  ^ , r f''SCEIndeed, one has the relation(3)________ P(G^q) = (-l^'^G; - g, -1).(*) This work is supported in part by the National Science Foundation grants DMR-9614170 (FYW and WTL) and DMS-9705779 (CK).Keywords: Graph colorings — Rooted Tutte polynomial — Planar partition — Dualityrelation.Math. classification: 05C30 - 05C15.1104 F.Y. WU, C. KING AND W.T. LUIn view of (3), it is useful to write (2) as(4) Q(G'^v) = v-^ ̂ (^ l̂,SCEso that for vt = q = positive integers, the Tutte polynomial (4) generatescolorings of components of spanning subgraphs of G with q colors and edgeweights z?, instead of v == —1.For planar G with dual graph Op, it is well-known that the Tuttepolynomial possesses the duality relation(5) vQ{G'^v)=tQ(GD\v.t),a relation first observed by Whitney [5].2. The rooted Tutte polynomial.We extend the definition (4) to a rooted Tutte polynomial.A vertex is rooted, or is a root, if it is colored with a prescribed (fixed)color. A graph is rooted if it contains rooted vertices.Let R denote a set of n roots located at vertices {r i , r2? • • • ^n}-A color configuration is a map x: R i—^ G, and as a convenient shorthand wewrite x(ri) = Xi for i == 1 ,2 , . . . , n. A component of a spanning subgraphis exterior if it contains one or more roots, and is interior otherwise. Anexterior component is proper if all roots in the component are of the samecolor. A spanning subgraph G'{S) is proper if all its exterior componentsare proper. An edge set Sx C E is proper if the spanning subgraph G'{Sx)it generates is proper.For a prescribed color configuration {a; 1 ,^2 , . . . ,^n} of the n roots,we introduce in analogy to (4) the rooted Tutte polynomial^ ̂(6) Qx^...x^G^v) = v-^ ̂  {vtr-^v^^S^CEwhere the summation is taken over all proper edge sets 63;, and pin (63;) isthe number of interior components of G^&c). Thus, as in (3), we have for(^ Strictly speaking, it is the expression v\v\Qx^x•2...xn(G\t^v) which is a polynomialin v and t.ON THE ROOTED TUTTE POLYNOMIAL 1105positive integral q the relation(7) (-l)I^^JG;-^-!)= the number of g-colorings of Gwith color configuration {x\^x^,..., Xn}'Clearly, the expression (6) depends on how the n roots are partitioned intosubsets of different colors, and the actual colors do not enter the picture.The coloring configuration {a;i, x^ , . . . , Xn} induces a partition X of Rinto blocks (subsets) such that all roots in one block are of one color, andcolors of different blocks are different. Namely, two elements r^rj € Rbelong to the same block of X if and only if they have the same prescribedcolor Xi = Xj.Consider now the summation in (6). Let G'{S) be any (not necessarilyproper) spanning subgraph of G. The connected components of G'(5)induce a partition on the set of vertices V of G. We get hence also apartition 7r(5') on the set of rooted vertices -R by restricting this partitionto R. Clearly, the spanning subgraph G'(Sx) is proper if and only if thepartition 7r(Sx) is a refinement of the partition X. It follow that we canrewrite (6) as(8) Qx{G'^v)= ̂  Fx'(G;^),x'^xwhere(9) Fx'{G^v)=v-^ ̂  {vtY-^v^.S^CETr(S^=X'Here, we have abbreviated Qxix2...xrz by Qx^ which is permitted since theactual colors do not enter the picture at this point. Also it is understoodthat G is now a rooted graph, with root set R.The expression (8) assumes the form of a transformation of a partiallyordered set. Its inverse is given by the Mobius inversion(10) Fx(G'^v) = ̂ ^X^X)Qx'(G'^v\x 'where (see [6])^^ ^-l^'-W n (^(X')-!)!, i fX^X,(11) /^(X^X) = < blocks GX10, otherwise,n^-^) being the number of blocks of X' that are contained in the block bof X. Note that for n == 1 we have \X\ = \X' = 1, pin(S^) = p(S^) - 1,1106 F.Y. WU, C. KING AND W.T. LUand all edge sets S C E are proper. Hence we have(12) Fx[G^ v) = Qx(G'^ v) = (vt^Q^ v), n = 1.This completes the definition and general description of the rooted Tuttepolynomial for any graph G.3. Planar graphs.From here on we consider G being planar with the n roots residingaround a single face of G. Without the loss of generality, we can choosethe face to be the infinite face and order the roots in the sequence{ r i , r 2 , . . . ,rn,r-t} as shown in Fig. 1.r2 rsFigure 1. A planar graph G with n roots. The graph is denotedby the shaded region and the n roots by the black circles.A partition X of the n roots is non-planar if two roots of one blockseparate two roots of another block in the cyclic sequence. Otherwise X isplanar. For a given n, there are bn partitions, where (see [9])00 00 00(13) bn= ̂  [n!/]^!)"1^!], ^i/m^=n,rn^=0 i/=l y=land of the bn partitions c- - ̂ ,are planar (see [7], [8]). We shall adopt the convention of writingX = { < 7 , H . . . , . . . }for colors {xi == xj, Xk = x^ = • • • , . . . } , with {ij}, {k£ . . . } , . . . each in order(see [8]). For example, two partitions for n = 5 areX^= {123,4,5}, |Xi|=3, planar,( J - O )X2 = {24,351}, |X2| = 2, non-planar.Now if G is planar and X' is non-planar then by definition the summandin (9) is empty and one has Fx'{G\t, v) == 0. Thus we haveON THE ROOTED TUTTE POLYNOMIAL 1107PROPOSITION 1. — For planar G(16) Fx(G\ t, v) =0, if X is non-planar.This proposition was first established in [9] for the Potts modelcorrelation function (see section 6) by considering its graphical expansionsimilar to the consideration given in the above.As a consequence of Proposition 1 and the use of (10), we now have:COROLLARY 1. — Rooted Tutte polynomials associated with non-planar partitions can be written as linear combinations of the rooted Tuttepolynomials associated with (refined) planar partitions.Corollary 1 leads to the sum-rule identities reported in [9] for thePotts correlation function. In the case of n = 4, for example, the identity^{13,24} (G;^)=0leads to the sum rule (see [9])(17) 0{13,24}(G;^) = Q{13,2,4}(G;^) +Q{1,3,24}(G;^)-Q{l,'2,3^}{G',t,v).From here on we shall restrict our considerations to rooted Tuttepolynomials associated with the Cn planar partitions only.4. The graph G*.The rooted Tutte polynomial (6) possesses a duality relation forplanar graphs, which relates the rooted Tutte polynomial on a graph G tothat of a related graph G*. Here we define G*. Starting from a planar G,place an extra vertex / in the infinite face and connect it to each root of Gby an edge. This gives a new graph G", which has one more vertex than Gand n additional edges. The dual graph of G" is also planar, and it hasa face F containing the extra vertex /. Now remove the n edges on theboundary of F^ and the resulting graph is G*.It is readily seen that the graph G* has(18) \V^\=\VD\+n-l1108 F.Y. WU, C. KING AND W.T. LUvertices where [Volis the number of vertices of GD, the dual of G, and thereis a one-one correspondence between the edges of G and G*. We denote theset of n vertices {r^, r^...,r^} of G* surrounding the face F by R*, withT^ residing between the two edges (/, r^-i) and (/, r^) of G", where r-o = r^.An example of a G and the related G* for n = 4 is shown in Fig. 2.r2x-r i Hrl T--3tX N -II---- - -13---?C-..(?---» ' ^'x''---y"'1- - -X-"^3'X'Figure 2. A graph G (solid lines) and the related graph G*(broken lines) for n = 4. Black circles denote roots of G andcrosses denote vertices of G*.Clearly, the relation of G to G* is reciprocal, namely, we have(G*)* = G.Now each planar partition X of R induces a partition X* of the set R*(see [10]). In order to define X*, for each block b of X we choose a pointin the infinite face of G, and connect all roots ri in b to this point bydrawing new edges. Because X is planar, the points for the blocks can bechosen so that the edges of different blocks do not cross, and the resultingextended graph is still planar. So this process divides the infinite face intoregions. The induced partition X* is then described by the condition thatall roots of R* in one region are regarded as belonging to one block of X*.Alternately, for another way of defining X*, let P^, 1 < i < j <: n, be apartition of R into two blocks, the sets { r ^ , . . . ,rj_i} and { r j , . . . ,7*z_i},where all numbers are modulo n. Then, the partition X* induced by X isdefined by the condition that the two roots r̂ * and rj belong to the sameblock in X* if and only if X is a refinement of P^.If X induces X*, we writeX X*.ON THE ROOTED TUTTE POLYNOMIAL 1109Clearly, X* is planar, and we have(19) |X|+ |X*|=n+l .For the planar partition Xi in (15), for example, we have(20) Xi = {123,4,5} -^ Xi* = {2,3,451}, |Xi| = |X^| == 3.However, as a result of our labelling convention, the color configuration ofthe partition (X*)* further induced by X* is a cyclic shift of that of X,namely,(21) {^ i ,^2 , . . . , ^n} —^ X* —^ {a;n,a;i,...,^-i}.In the example above, for instance, we have(22) {123,4,5} —— {2,3,451} —— {234,5,1}.Finally, there is a one-one correspondence between the edge sets Eof G and £'* of G*, an edge set S C E defines a "complement" edge set5* C E* by the condition that an edge is included in S* if and only if itscorresponding edge is not included in S. Clearly, we have(5*)* == S.5. The duality relation.The rooted Tutte polynomial arises in statistical physics as thecorrelation function of the Potts model (see next section). In a recentpaper [10] we have established a duality relation for the Potts correlationfunction for planar G. However, the proof of the duality relation givenin [10] is cumbersome and not easily deciphered in graphical terms. Here were-state the results as two propositions in the context of the rooted Tuttepolynomial, and present direct graph-theoretical proofs of the propositions.PROPOSITION 2. — For planar G and G* and the associated planarpartitions X —> X*, we have(23) v^Fx(G^v)=t^\Fx^G^^^.1110 F.Y. WU, C. KING AND W.T. LUProof. — Let Sx be a proper edge set on G. We have the Euler relation(24) |5,|+|^|=|V|+|l^|-2and, after eliminating n and \VD\ using (18), (19) and (24), the identity(25) |^|+|x|-|y|=|^|+|x*|-|y*|,which holds for any proper edge set Sx. Note that we have also the fact(26) Tr(^) = X if and only if 7r(S^) = X\Let c(S^) the number of independent circuits in the spanning sub-graph G'(5^). Then we have(27) p(^)=c(^*)+|X|.Also, starting from the \V*\ isolated vertices on G*, one constructs G'^S^)by drawing edges of S^ on G* one at a time. Since each edge reduces thenumber of components by one except when the adding of an edge completesan independent circuit, one has also(28) p(S^=\Vf\-\S^+c^).Eliminating c(5^) using (27) and (28) and making use of the relationsfP(^)=Pin(^)+|X|,(29) <\p(S^=p^(S^+\X^one obtains(30) pin(^) =Pin(^) + m - |̂ | - |X*1.Proposition 2 now follows from the substitution of (30) into the right-handside of (23) where, explicitly,(31) Fx'(Gt•,v,t)=t-\v'\ ̂  (^••^W,S^CE'^(S:)=X"and the use of the identities (25) and (26). This completes the proof ofProposition 2.ON THE ROOTED TUTTE POLYNOMIAL 1111Proposition 2 was first conjectured in [9] and established later in [10]in the context of Potts correlation functions (see next section) without theexplicit reference to the polynomial form (9).Remark. — For n == 1, the duality relation (23) for the rooted Tuttepolynomial becomes the duality relation (5) for the Tutte polynomial. Thisis a consequence of (12).PROPOSITION 3.1) The rooted Tutte polynomials associated with the Cn planarpartitions for G and G* are related by the duality transformation(32) Ox(G;^)=^r,(X,y)Qy(G*;^),ywhere Tn is a Cn x Cn matrix with elements(33) T,(X, Y) = r+1 ^ (^-i^i/^ v'). x' -^ y.x'-^x2) The matrix Tn satisfies the identity(34) [Tn}\X, X') = 6(x^x^6(x^ ̂ )... 6{x^ x[).Proof. — The transformation (32) follows by combining (8) and (10)with Proposition 2, and its uniqueness is ensured by the uniqueness of theMobius inversion. The property (34) is a consequence of (21). DProposition 3 was first given in [10] in the context of the Pottscorrelation function (see next section). Explicit expression of Tn forn = 2,3,4 can be found in [10] and [11].6. The Potts and the random cluster models.It is well-known in statistical physics that the Tutte polynomial givesrise to the partition function of the Potts model (see [12]). In view of theprominent role played by the Potts model in many fields in physics, it isuseful to review this equivalence and the further equivalence of the rootedTutte polynomial with the Potts correlation function.1112 F.Y. WU, C. KING AND W.T. LUThe estate Potts model [13] is a spin model defined on a graph G.The spin model consists of \V\ spins placed at the vertices of G with eachspin taking on q different states and interacting with spins connected byedges. Without going into details of the physics [12] which lead to the Pottsmodel, it suffices for our purposes to define the Potts partition function(35) Z(G^v)== ̂ (f^l5!,SCEthe ?7-point partial partition function(36) Zx(G'^v)= ̂  ̂ in(̂ l̂S^CEand the n-point correlation function(37) Pn(G;^i, x^... ,xn) = Pn(G;X) = Zx(G;g, v)/Z(G;q, v),where again, in analogy to notation in Sections 1 and 2, we have denotedthe color configuration {x^, x ^ , . . . , Xn} by the associated partition X. Moregenerally, for any real or complex g, the partition function (35) defines therandom cluster model of Fortuin and Kasteleyn [14], which coincides withthe Potts model for integral q.Relating this to the Tutte polynomial, we now have(Z{G;q,v)=v^Q(G;t,v),(38) ^ Zx(G;q,v) = v^Qx(G;t,v),( Pn(G;X) = Qx(G;t,v)/Q(G;t,v),for q = vt. The duality relation (5) for the Tutte polynomial then impliesthe following duality relation for the Potts partition function [10], [13], [15](39) vl-^Z(G•,q, v) = (^)l-lv°l^(Gp;g, v*),where(40) vv* = q.One further defines the dual correlation function(41) PW;X^ = q Z^G^,q^)/Z(GD\qX).and also the functions Ax and Bx' by(42) Pn(G;X)= ̂  Ax'(G;q,v)x'^xON THE ROOTED TUTTE POLYNOMIAL 1113and(43) P,*(G*;X*)= ^ Bx-(G*;^*).x'-^x-Then, Proposition 2 leads to the relation(44) Ax(G;g,^) = g-WBx^G*;^*), X -. X\which is the main result of [10].7. Summary and discussions.We have introduced the rooted Tutte polynomial (6) as a two-variablepolynomial associated with a rooted graph and deduced a number ofpertinent results.Our first result is that the rooted Tutte polynomial assumes theform (8) of a partially order set for which the inverse can be uniquelydetermined. For planar graphs and all roots residing surrounding a singleface, we showed that (Proposition 1) the inverse function vanishes for non-planar partitions of the roots. We further showed that the inverse functionsatisfies the duality relation (23) (Proposition 2) which, in turn, leads tothe duality (32) for the rooted Tutte polynomial (Proposition 3). We alsoreviewed the connection of the Tutte and rooted Tutte polynomials withthe Potts model in statistical physics.Finally, we remark that results reported here have previously beenobtained in [9] and [10] in the context of the Potts correlation function. Here,the results are reformulated as properties of the rooted Tutte polynomialand thereby permitting graph-theoretical proofs.Noted added. — The polynomial Q(G;x, y) is now commonly referredto as the dichromatic polynomial and the dichromatex(G;:r, y) = (x - ̂ Q^x -1^-1)is now commonly known as the Tutte polynomial. We are indebted toProfessor W.T. Tutte for this remark.Acknowledgments. — We are grateful to the referee for providing anindependent proof of Proposition 1 and numerous suggested improvementson an earlier version of this paper.1114 F.Y. WU, C. KING AND W.T. LUBIBLIOGRAPHY[I] G.D. BIRKHOFF, A determinant formula for the number of ways of coloring of amap, Ann. Math., 14 (1912), 42-46.[2] W.T. TUTTE, A contribution to the theory of chromatic polynomials, Can. J.Math., 6 (1954), 80-91.[3] W.T. TUTTE, On dichromatic polynomials, J. Comb. Theory, 2 (1967), 301-320.[4] W.T. TUTTE, Graph Theory, in Encyclopedia of Mathematics and Its Appli-cations, Vol. 21, Addison-Wesley, Reading, Massachusetts, 1984, Chap. 9.[5] H. WHITNEY, The coloring of graphs, Ann. Math., 33 (1932), 688-718.[6] See, for example, L.J. VAN LINT and R.M. WILSON, A course in combinatorics,Cambridge University Press, Cambridge, 1992, p. 301.[7] H.N.V. TEMPERLEY and E.H. LlEB, Relations between the percolation andcolouring problem and other graph—theoretical problems associated with regularplanar lattice: some exact results for the percolation problem, Proc. Royal Soc.London A, 322 (1971), 251-280.[8] W.T. TUTTE, The matrix of chromatic joins, J. Comb. Theory B, 57 (1993),269-288.[9] F.Y. WU and H.Y. HUANG, Sum rule identities and the duality relation forthe Potts n— point boundary correlation function, Phys. Rev. Lett., 79 (1997),4954-4957.[10] W.T. LU and F.Y. WU, On the duality relation for correlation functions of thePotts model, J. Phys. A: Math. Gen., 31 (1998), 2823-2836.[II] F.Y. WU, Duality relations for Potts correlation functions, Phys. Letters A, 228(1997), 43-47.[12] See, for example, F.Y. WU, The Potts Model, Rev. Mod. Phys., 54 (1982),235-268.[13] R.B. POTTS, Some generalized order-disorder transformations, Proc. Camb.Philos. Soc., 48 (1954), 106-109.[14] C.M. FORTUIN and P.W. KASTELEYN, On the random-cluster model I. Intro-duction and relation to other models, Physica, 57 (1972), 536-564.[15] F.Y. WU and Y.K. WANG, Duality transformation in a many-component spinmodel, J. Math. Phys., 17 (1976), 439-440.F.Y. WU & W.T. LU,Northeastern UniversityDepartment of PhysicsBoston, Massachusetts 02115 (USA).fywu@neu.eduwtlu@albert.physics.neu.edu&C. KING,Northeastern UniversityDepartment of MathematicsBoston, Massachusetts 02115 (USA).king@neu.edu

https://aif.centre-mersenne.org/item/10.5802/aif.1709.pdf

On the rooted Tutte polynomial

Abstract

Similar works

Full text

Available Versions

Numérisation de Documents Anciens Mathématiques

Crossref

Annales de l’institut Fourier (AIF)