Medial Graphs and the Penrose Polynomial. Joanna A. Ellis-Monaghan Department of Mathematics and Statistics University of Vermont Burlington, VT 05405

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1 Medial Graphs and the Penrose Polynoial Joanna. Ellis-Monaghan Departent of Matheatis and Statistis University of Veront Burlington, VT Irasea Sariento Departaento de Mateátias CINVESTV v. IPN 2508 CP 07000, Méxio D. F. bstrat We onstrut a one variable graph polynoial, N( G, W; x ), on the spae of planar drawings of Eulerian graphs with axdeg = 4 and weight syste W. When G is the edial graph of a planar graph G, with an appropriate weight syste W, then N( G, W ; x) = P( G ; x), where P( G; x) is the Penrose polynoial of G. We give a siple obinatorial N G, W; x, whih has as a speial proof of an identity for ( ) ase an identity very siilar to one derived using Hopf algebrai tehniques in [Sar]. This identity then gives relations for the Penrose polynoial of a graph in ters of the Eulerian subgraphs of its edial graph. Introdution The Penrose polynoial of a graph, defined ipliitly by Roger Penrose in ppliations of Negative Diensional Tensors, [Pen69], enodes edge oloring inforation of planar graphs. This polynoial an be oputed via weighted skein, or transition, relations in its edial graph. In this paper we onstrut a new polynoial defined on drawings of graphs in the plane with speified weights assigned to eah vertex state. Properties of this polynoial are then used to get an identity for the Penrose polynoial. The following onventions are used through this paper. Graphs ay have loops and ultiple edges. graph is said to be Eulerian if all its verties have even degrees, but onnetedness is not required. yle is a graph isoorphi Ellis-Monaghan & Sariento page 1 of 12 Penrose Polynoial 5/12/01

2 to a polygon. The notation k(g) will be used to indiate the nuber of oponents of G, not ounting isolated verties. Reall that the edial graph of a onneted planar graph G is onstruted by putting a vertex on eah edge of G and drawing edges around the faes of G. Speifially, two verties of the edial graph of G are joined by an edge if the orresponding edges in G are neighbors in the yli order of edges around a vertex. For exaple: The graph G The edial graph of G vertex state is a hoie of loal reonfiguration of a graph at a vertex by pairing the edges inident with that vertex. In this paper we will only onsider those vertex states where all the edges are paired (no singleton edges). For exaple, the three possible vertex states of a vertex of degree 4 are: graph state is the result of hoosing an allowed vertex state (different appliations ay use only speifi vertex states) at eah vertex of degree greater than 2. For exaple, the three possible graph states of this graph are: We will write St ( G) for the set of graph states of G. Ellis-Monaghan & Sariento page 2 of 12 Penrose Polynoial 5/12/01

3 skein-relation for graphs is a foral su of possibly weighted vertex states, together with an evaluation of the terinal fors (the graph states, typially a disjoint union of graphs with axdeg=2). See [E-M98] for a detailed disussion of these onepts, whih are appropriated fro knot theory, in their ost general for. skein-type, or transition, polynoial is one whih is oputed by repeated appliations of skein relations. See [Jae90] for a oprehensive treatent of these in the ase of 4-regular graphs. Definition 1: weight syste W on a planar drawing of an Eulerian graph with axdeg=4 is an assoiation of a value (a vertex state weight) to eah state at eah vertex of degree 4 in G. These values ay in general be in any ring, but here they will typially be integers. Thus there are three values assoiated to eah vertex of degree 4 in G. Definition 2: The state weight of a graph state S of a graph G with weight syste W is ω( S) = ω( S, v), where ( Sv, ) ω is the vertex state weight of the vertex state at v in the graph state S, and where the produt is over all verties of deg 4 in G. The Penrose Polynoial P( G; x ) The Penrose polynoial of a planar graph was defined ipliitly in [Pen69], and an exellent exposition an be found in [ig97]. It an be oputed via skein relations applied to its edial graph (see [Jae90] for exaple), and we will use this approah here. Let G be a planar graph, and let G be its edial graph, two-olored with the unbounded fae olored white. Then P( G; x) an be oputed by applying the following skein-relation to the verties of degree 4 in G : v = - This redues the graph to a foral su of unions of irles (yles with one edge and no verties), whih are then eah evaluated to x: = x. Ellis-Monaghan & Sariento page 3 of 12 Penrose Polynoial 5/12/01

4 The Penrose polynoial also has the following state odel forulation: r (( ) ( S ) k S) PG ( ; x) = ( Sv, ) x = 1 x k ( S ω ) ( ), St ( G) v G St ( G) where ω ( S, v) is + 1 for the white non-rossing state at v, 0 for the blak nonrossing state, and 1 for the rossing state, ks ( ) is the nuber of oponents in the graph state S, and r( S ) = the nuber of rossing vertex states hosen in =. v G For exaple: the state S, so that ω r (, ) ( 1) ( S Sv ) G = G = -x +x -x 2 +x +x -x 2 +x 3 -x 2 PGx ; = x 3x+ 2x. Thus, ( ) 3 2 Ellis-Monaghan & Sariento page 4 of 12 Penrose Polynoial 5/12/01

5 The Penrose polynoial has soe surprising properties whih ake its study so entiing, partiularly with respet to the faous Four Color Theore. The Four Color Theore, whih states that the regions of every planar bridgeless graph an be properly olored with four olors, an be shown to be equivalent to showing that every planar, ubi, onneted graph an be properly edge-olored with three olors. The Penrose polynoial, when applied to planar, ubi, onneted graphs, enodes exatly this inforation (see [Pen69]): 1 = = # edge-3-olorings of G. 4 2 ( ;3) PG ( ; 2) PG The polynoial N( G; x ) V We now define a new graph polynoial whih will have the property that N( G, W ; x) = P( G ; x) whenever G is a planar graph and an appropriate weight syste W is assigned to G. Thus studying this new polynoial will lead to insights into the Penrose polynoial. Let G be an Eulerian ultigraph (loops and ultiple edges allowed) with ax deg = 4, drawn in the plane siply (only two edge rossing at any given point), and with the regions two-olored with the unbounded region olored white. Identify two graphs whih are abient isotopi to eah other, or whih an be transfored one into the other by swithing whih of two edges rosses over the other, or whih are the sae if isolated verties are ignored. Write G,W for suh a graph drawing with an assoiated weight syste W. Let Γ= span {all suh G, W under this equivalene}. C Reursive Definition: Define N : Γ C [] x reursively by repeatedly applying the following skein relation at any vertex of degree 4, and then evaluating the terinal fors (the graph states) by identifying eah yle with the variable x. Here, α () v, β () v and γ () v are the vertex state weights assigned to eah state at v by the weight syste W. v = α(v) + β(v) +γ(v) Ellis-Monaghan & Sariento page 5 of 12 Penrose Polynoial 5/12/01

6 = x. State Model Definition: NGWx (, ; ) = Sv, x = S x St( G) St( G) k( S) k( S) ω( ) ω( ). The polynoial NGWx (, ; ) is essentially the sae as the transition polynoial Q( GW,, x ) in [Jae90]. They differ by a fator of x, and here we retain verties of deg 2 in the reursion while they are elided in [Jae90]. Note that sine this state odel definition is learly equivalent to the reursive definition, and is independent of the order of the verties to whih the reursion is applied, NGWx (, ; ) is well-defined. lso, sine the states, both vertex and graph, are in ters of loal reonfigurations, NGWx (, ; ) is welldefined with respet to the equivalene relations within the definition of Γ. Beause the assignent of vertex state weights ay depend on the drawing, it is ertainly possible that N ay depend on the partiular drawing of a graph. For exaple, onsider the following two drawings of the graph with one vertex and two edges. Deterine the weight systes by two-oloring the regions with the outer region olored white, and apply the sae weights as for the Penrose polynoial (1 for white non-rossing, 0 for blak non-rossing, and 1 for rossing). Let L be the drawing on the left, and R be the drawing on the right. 2 N L, W, x = 0, but N( R, W, x) = x x. Then ( ) However, extending the lass of graphs on whih N is defined fro planar graphs to graph drawings is neessitated by the fat that a hoie of vertex states in the reursion relation ay transfor a planar graph into a non-planar graph drawn in the plane. This an be seen in the following exaple, where for the rossing state at v, the resulting graph is hoeoorphi to K 5. Ellis-Monaghan & Sariento page 6 of 12 Penrose Polynoial 5/12/01

7 v Generating Funtion Definition: Colleting like ters in the preeding state odel definition leads to the N G, W; x. following generating funtion for of ( ) (, ; ) ( ) n = n, N G W x f G x where f ( G) ( S) n oponents. = ω, where the su is over all states of G with n LEMM 3 Let G be the edial graph of a planar graph G. Give G a weight syste W by two-fae-oloring G (unbounded region olored white), and assigning a value of +1 for the white non-rossing state at eah vertex v, 0 for the blak non-rossing state, and 1 for the rossing state. Then N( G, W ; x) = P( G ; x). Proof: With this weight syste, N( G ), ; W x is exatly the sae as the state odel definition for the Penrose polynoial, exept that the graph states here retain inforation about the verties of degree 2. However, sine yles of all lengths all evaluate to x, the end result is the sae. Furtherore, sine the edial graphs of any two abient isotopies of a onneted planar graph are abient isotopi, it follows that N( G W x) P( G x) /// The strutural properties of N( G; x )., ; ; = is onsistent as well. We now turn our attention to the strutural properties of N( G, W; x ). Let GH be the disjoint union of two graphs G and H drawn in the plane with weight systes W and W, respetively. Sine the skein relation an be applied first to N G, W; x all the degree 4 verties in G, and then to those in H, it follows that ( ) Ellis-Monaghan & Sariento page 7 of 12 Penrose Polynoial 5/12/01

8 is ultipliative on disjoint unions of graphs in Γ, so that N( GHW, "; x) = N( GW, ; x) N( HW, '; x), where W is the weight syste whih assigns to v VGH ( ) the weights it originally reeived fro W (or W ) as a vertex of G (or H). Thus N : Γ C [] x is an algebra ap. Theore 7 below will have as a orollary that N : Γ C [] x is also a oalgebra ap. Definition 4: Let be an Eulerian subgraph of a graph G in Γ. Let V ( / ) 2 G be the verties of degree 4 in G that have degree 2 when restrited to, and let V ( / ) 4 G be those that have degree 4. When there is no danger of onfusion about the underlying graph G, we will write Vi ( ) for Vi ( / G ). Definition 5: Let be an Eulerian subgraph of a graph G in Γ, and let v V ( / G) 2. Then there are three different possible onfigurations for in G at v, and they orrespond to the three possible vertex states at v. (The heavy edges are in.) These are the three different transitions of in G. v v v Definition 6: σ( / G) ω(, v) =, where the produt is over V ( / G ). 2 Here ω ( v, ) is the weight of the vertex state at v whih orresponds to the transition of in G. When there is no danger of onfusion, σ ( ) will be written for σ ( / G). THEOREM 7 N( G, W; x y) σ ( ) N(, W ; x) N(, W ; y) the su is over all Eulerian subgraphs of G. + =, where Ellis-Monaghan & Sariento page 8 of 12 Penrose Polynoial 5/12/01

9 Here is written for G, the restrition of G to (as a drawing). The weight syste for, denoted W, is inherited fro W in suh a way that if v V4 ( ), then the state weights at v as a vertex of are the sae as they were as for v as a vertex of G. This holds siilarly for. Proof: By the state odel definition, k S ks () NGWx (, ; + y) = S x+ y = S xy St( G) St( G) r= 0 r ( )( ) ( ) ( ) k( S) r k( S) r ω ω. Thus, the oeffiient of a b x y on the left-hand-side is: ω ( S ) a + b a, where the su is over S St( G) suh that ( ) k S = a+ b. By the sae state odel definition, the right-hand-side beoes k S k( S ) σ ω S x ω S y ( ) ( ), ( ) ( ) where the outer su is over all Eulerian subgraphs of G, the first inner su is over all graph states S of, and the seond inner su is over all graph states S of. Thus, the oeffiient of a b x y on the right-hand-side is: σ( ) ω( S ) ω( S ), where the outer su is over all Eulerian subgraphs of G, the first inner su is over all graph states S of with a oponents, and the seond inner su is over all graph states S of with b oponents. Now onsider a state S of G with a + b oponents. Choose a of those oponents to oprise S. Thus onsists of the edges in those a oponents and hene is Eulerian. The state S is deterined by the state S, as is onsequently the state S whih has b oponents. Note that there are Ellis-Monaghan & Sariento page 9 of 12 Penrose Polynoial 5/12/01

10 a+ b a ways to do this. Thus the oeffiients of a b x y are equal on both sides of the equation if and only if ω( S) σ( ) ω( S ) ω( S ) However, notie that ω( S) ω( S, v) =. = = ω( v, ) ω( S) ω S = σ ω S ω S. ( ) v V ( ) v V ( ) v V ( ) ( ) ( ) ( ) Thus this is true, whih opletes the proof. /// COROLLRY 8: N G W x x N W x N W x. (, ;1 + 1 ) = σ ( ) (, ; ) (, ; ) Proof: This follows iediately fro letting x= 1 x and y = x 1. /// This now eans that N : Γ C [] x is a oalgebra ap, where C[ x] is given the struture of the binoial bialgebra, and Γ has a oultipliation given by Δ ( GW, ) = σ ( )( W, ) (, W ), where the su is over all Eulerian reursively by ( ) ( ) subgraphs of G with weight systes inherited fro W. It is routine to hek that Γ is in fat a Hopf algebra with antipode given ζ GW, = σ ζ ( ) where the su is over all nonepty Eulerian subgraphs, and where ζ ( ) =1 for, any graph with no edges. Thus, sine N is a bialgebra ap, it is autoatially a Hopf ap, i.e. respets the antipode (see [Swe, lea 4.04]). In partiular, we get the following lea: LEMM 9 N( G, W; x) = N( ζ ( G, W) ; x). Proof: Sine the antipode of [ x] a Hopf ap, this follows iediately. /// ζ =, and N is C is given by ( px ( )) p( x) s an iportant speial ase, onsider the hereditary set given by graphs in Γ where the weights in the weight systes are restrited to {-1, 0, 1}. This is a hereditary set sine the substrutures (Eulerian subgraphs with inherited weight systes) reain in the set. Beause it is a hereditary set, it fors a sub-hopf Ellis-Monaghan & Sariento page 10 of 12 Penrose Polynoial 5/12/01

11 algebra of Γ (see [Sh94, Sh95] for ore inforation about hereditary sets and Hopf algebras). By onsidering the oultipiation and antipode of this sub-hopf algebra together with Lea 3, we get the follow two results, whih are equivalent to those in [Sar]. THEOREM 10 [SR, THEOREM 4.3] Let G be the edial graph of a planar graph G. Give G a weight syste W by two-fae-oloring G (unbounded region olored white), and assigning a value of +1 for the white non-rossing state at eah vertex v, 0 for the blak non-rossing state, and 1 for the rossing state. Then P( G, x y) σ ( ) N( W, ; x) N(, W ; y) + = where the su is over all Eulerian subgraphs of G with weight systes inherited fro W. Proof: This follows iediately fro Lea 3 and theore 7. THEOREM 11 [SR, THEOREM 5.2] Let G be the edial graph of a planar graph G. Give G a weight syste W by two-fae-oloring (unbounded region olored white), and assigning a value of +1 for the white non-rossing state at eah vertex v, 0 for the blak non-rossing state, and 1 for the rossing state. Then P( G, x) N( S( G, W) ; x) =. Proof: This follows iediately fro leas 3 and 9. /// This theore is espeially useful beause it an be used to give interpretations for the Penrose polynoial at negative values (see [Sar]). G Bibliography [ig97] M. IGNER, The Penrose polynoial of a plane graph, Math. nn., 307 (1997) [E-M99] J.. ELLIS-MONGHN, New Results for the Martin Polynoial, Journal of Cobinatorial Theory, Series B, 74 (1998) [Jae90] F. JEGER, On Transition Polynoials of 4-regular Graphs, Cyles and Rays (Montreal, PQ, 1987) , NTO dv. Si. Inst. Ser. C: Math. Phys. Si., 301, Kluwer ad. Publ., Dordreht, [Pen69] R. PENROSE, ppliations of Negative Diensional Tensors, Cobinatorial Matheatis and its ppliations; Proeedings of a onferene held at Ellis-Monaghan & Sariento page 11 of 12 Penrose Polynoial 5/12/01

12 [Sar] [Sh94] the Matheatial Institue, Oxford, 1969, London, New York, adei Press, (1971), I. SRMIENTO, Hopf lgebras and the Penrose Polynoial, preprint. W. SCHMITT, Inidene Hopf algebras, Journal of Pure and pplied lgebra, 96 (1994) [Sh95] W. SCHMITT, Hopf lgebra Methods in Graph Theory, Journal of Pure and pplied lgebra, 101 (1995) [Swe69] M. E. SWEEDLER, Hopf lgebras, New York: W.. Benjain, In., Ellis-Monaghan & Sariento page 12 of 12 Penrose Polynoial 5/12/01