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Semisymmetric Graphs from Polytopes

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Semisymmetric Graphs from Polytopes Semisymmetric Graphs from Polytopes Barry Monson¤ University of New Brunswick Fredericton, New Brunswick, Canada E3B 5A3 ([email protected]) Toma·z Pisanskiy University of Ljubljana, FMF Jadranska 19, Ljubljana 1111, Slovenia; and Faculty of Education, University of Primorska, Cankarjeva 5, Koper 6000, Slovenia ([email protected]) Egon Schultez Northeastern University Boston MA 02115, USA ([email protected]) Asia Ivi¶c Weissx York University Toronto, Ontario, Canada M3J 1P3 ([email protected]) Abstract Every ¯nite, self-dual, regular (or chiral) 4-polytope of type f3;q;3g has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by drop- ping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope. Key Words: semisymmetric graphs, abstract regular and chiral polytopes. AMS Subject Classi¯cation (1991): Primary: 05C25. Secondary: 51M20. ¤Supported by the NSERC of Canada, grant 4818 ySupported by Ministry of Higher Education, Science and Technolgy of Slovenia grants P1-0294,J1-6062,L1-7230. zSupported by NSA-grant H98230-05-1-0027 xSupported by the NSERC of Canada, grant 8857 1 1 Introduction The theory of symmetric trivalent graphs and the theory of regular polytopes are each abundant sources of beautiful mathematical ideas. In [22], two of the authors established some general and unexpected connections between the two subjects, building upon a rich variety of examples appearing in the literature (see [4], [7], [10], [11], [28] and [29]). Here we develop these connections a little further, with speci¯c focus on semisymmetric graphs. In particular, we reexamine the Gray graph, described in [2, 3] and [18, 24], and here appearing as the medial layer graph of an abstract regular 4-polytope. We begin with some basic ideas concerning symmetric graphs [1, ch. 18-19]. Although some of the following results generalize to graphs of higher valency, for brevity we shall assume outright that G is a simple, ¯nite, connected trivalent graph (so that each vertex has valency 3). By a t-arc in G we mean a list of vertices [v]=[v0;v1;:::;vt] such that fvi¡1;vig is an edge for 1 · i · t, but no vi¡1 = vi+1. Tutte has shown that there exists a maximal value of t such that the automorphism group Aut(G) is transitive on t-arcs. We say that G is t-transitive if Aut(G) is transitive on t-arcs, but not on (t + 1)-arcs in G, for some t ¸ 1. Tutte also proved the remarkable result that t · 5 ([1, Th.18.6]). Any such arc-transitive graph is said to be symmetric. Each ¯xed t-arc [v]inat-transitive graph G has stabilizer sequence Aut G⊃Bt ⊃ Bt¡1 ⊃ :::⊃ B1 ⊃ B0; where the subgroup Bj is the pointwise stabilizer of fv0;:::;vt¡jg. Since Aut(G) is transitive on r-arcs, for r · t, the subgroup Bj is conjugate to that obtained from any other t-arc. In particular, Bt is the vertex stabilizer, whereas B0 is the pointwise stabilizer of the whole arc. In fact, B0 = f²g is trivial ([1, Prop. 18.1]), so that Aut(G) acts sharply transitively on t-arcs. (k) Each t-arc [v] has two successors, t-arcs of the form [v ]:=[v1;:::;vt;yk], where vt¡1;y1;y2 are the vertices adjacent to vt. The shunt ¿k is the (unique) (k) automorphism of G such that [v]¿k =[v ]. Also let ® be the unique automor- phism which reverses the basic t-arc [v]. Then ® has period 2 and ®¿1® equals ¡1 ¡1 + ¡ either ¿1 or ¿2 . We shall say that G is of type t or t , respectively. We can now assemble several beautiful results concerning Aut(G) (see [1, ch. 18]). Theorem 1 Suppose G is a ¯nite connected t-transitive trivalent graph, with 1 · t, and suppose G has N vertices. Then j t¡1 (a) For 0 · j · t ¡ 1 we have jBjj =2 . Also, jBtj =3¢ 2 and jAut(G)j =3¢ N ¢ 2t¡1 : (b) The stabilizers Bj are determined up to isomorphism by t : 2 t B1 B2 B3 B4 B5 1 Z3 2 Z2 S3 2 3 Z2 (Z2) D12 2 4 Z2 (Z2) D8 S4 2 3 5 Z2 (Z2) (Z2) D8 £ Z2 S4 £ Z2 (c) G is one of 7 types: 1¡; 2+; 2¡; 3+; 4+; 4¡ or 5+. (Here Zk is the cyclic group of order k, D2k is the dihedral group of order 2k, Sk is the symmetric group of degree k.) Useful lists of symmetric trivalent graphs appear in [4] and [7]. We refer to [22] for a description of several interesting examples. We now briefly describe some key properties of abstract regular and chiral polytopes, referring again to [22] for a short discussion, and to [23, 25, 26] for details. An (abstract) n-polytope P is a partially ordered set with a strictly monotone rank function having range f¡1; 0;:::;ng. An element F 2Pwith rank(F )=j is called a j-face; typically Fj will indicate a j-face; and P has a unique least face F¡1 and unique greatest face Fn. Each maximal chain or flag in P must contain n + 2 faces. Next, P must satisfy a homogeneity property : whenever F<Gwith rank(F )=j ¡ 1 and rank(G)=j + 1, there are exactly two j-faces H with F<H<G, just as happens for convex n-polytopes. It follows that for 0 · j · n ¡ 1 and any flag ©, there exists a unique adjacent flag ©j, di®ering from © in just the rank j face. With this notion of adjacency the flags of P form a flag graph (not to be confused with the medial layer graphs appearing below). The ¯nal de¯ning property of P is that it should be strongly flag–connected. This means that the flag graph for each section is connected. Whenever F · G are faces of ranks j · k in P, the section G=F := fH 2PjF · H · Gg is thus in its own right a (k ¡ j ¡ 1)-polytope. Since our main concern is with 4-polytopes, we now tailor our discussion to that case. A (rank 4) polytope P is equivelar of type fp1;p2;p3g if, for j =1; 2; 3, whenever F and G are incident faces of P with rank(F )=j ¡ 2 and rank(G)=j + 1, then the rank 2 section G=F has the structure of a pj- gon (independent of choice of F<G). Thus, each 2-face (polygon) of P is isomorphic to a p1-gon, and there are p3 of these arranged around each 1-face (edge) of P; and in every 3-face (facet) of P, each 0-face is surrounded by an alternating cycle of p2 edges and p2 polygons. The automorphism group Aut(P) consists of all order preserving bijections on P.IfP also admits a duality (order reversing bijection), then P is said to be self-dual; clearly Aut(P) then has index 2 in the group D(P) of all automor- phisms and dualities. (Note that D(P) = Aut(P) when P is not self-dual.) If P is self-dual and equivelar, then it has type fp1;p2;p1g. De¯nition 1 Let P be a 4-polytope. The associated medial layer graph G(P), or briefly G, is the simple graph whose vertex set is comprised of all 1-faces and 2-faces in P, two such taken to be adjacent when incident in P. 3 Remarks: Any medial layer graph G is easily seen to be bipartite and con- nected. Note that the more desirable phrase `medial graph' already has a some- what di®erent meaning in the literature on topological graph theory. To further focus our investigations, we henceforth assume that P is equivelar of type f3;q;3g, where the integer q ¸ 2. Thus G is trivalent, with vertices of two types occuring alternately along cycles of length 2q. We say that a t-arc in G is of type 1 (resp. type 2 ) if its initial vertex is a 1-face (resp. 2-face) of P. The fact that certain polygonal sections of P are triangular immediately implies that the action of D(P)onG is faithful, so that we may regard D(P), or Aut(P), as a subgroup of Aut(G) (see [22, x 2]). In Figure 1 we show a fragment of a polytope P of type f3; 6; 3g. The vertices of G are here represented as black and white discs, and the edges of G are indicated by heavy lines. w4 s w 3 z v11 := F v 3 v w 4 2 v := F v2 22 F3 v1 F0 v−1 w y v 1 0 w x 0 Figure 1: A fragment of a polytope of type f3;q;3g, with q =6. Since we shall soon assume that P is quite symmetric, it is useful now to ¯x a base flag ©=fF¡1;F0;F1;F2;F3;F4g in P. Given this, it is convenient to de¯ne v1 := F1;v2 := F2, and in general let v0 = v2q;v1;v2;:::;v2q¡1 = v¡1 denote alternate edges and polygons in the rank 2 section F3=F0 of P. Thus each vj is adjacent in G to vj§1, taking subscripts mod 2q. We also let wj be the third vertex adjacent to vj in G,as indicated in Figure 1.
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