full-symmetric embeddings of graphs on closed

FULL-SYMMETRIC EMBEDDINGS OF GRAPHS ON CLOSED SURFACES Atsuhiro NAKAMOTO∗ and Seiya NEGAMI†

Abstract Every 2-cell embedding of a graph G on a closed surface F 2 admits a group action of autohomeomorphisms of F 2 of order at most 4 E(G) . The embedding of G is full-symmetric if G attains the upper bound for the order. We shall discuss on such full-symmetric embeddings and classify those on the projective plane, the torus and the Klein bottle.

1. Introduction Weinberg [7] has shown an upper bound for the order of the automorphism group Aut(G) of a 3connected planar graph G: Aut(G) ≤ 4 E(G) Furthermore, the equality holds if and only if G is isomorphic to one of the platonic graphs, that is, the 1-skeletons of regular polyhedra. Since every 3-connected planar graph is uniquely and faithfully embeddable on the sphere, Aut(G) can be realized as the symmetry group of an embedding of G on the sphere and hence Aut(G) coincides with the order of symmetries of an embedding of G on the sphere. (See [4] for the uniqueness and the faithfulness of embeddings.) Extending this observation, we shall discuss those graphs embedded on closed surfaces that attain such an upper bound. Let G be a graph embedded on a closed surface F 2 and let V (G), E(G) and F (G) denote the sets of vertices, edges and faces, respectively. Let Γ be the group consisting of all the auto-homeomorphisms h : F 2 → F 2 which induce automorphisms of G with h(G) = G. Two homeomorphisms h and h′ ∈ Γ are said to be equivalent, written by h ∼ h′ , if they act on the triple (V (G), E(G), F (G)) in the same way. The quotient group Γ/∼ is called the symmetry group of G and is denoted by Sym(G). If G has a vertex of degree at least 3, then Sym(G) can be regarded as a subgroup in Aut(G). Suppose that G is 2-cell embedded on F 2 . A triple {v, e, A} of a vertex v ∈ V (G), an edge e ∈ E(G) and a face A ∈ F (G) is called a flag (associated with e) if v, e and A are mutually incident, that is, e lies along the boundary of A and v is one of the ends of e. Such a triple corresponds to the triangle region with v, the middle point of e and the central point of A as its vertices. Two auto-homeomorphisms of F 2 induce the same element of Sym(G) if a fixed flag is carried onto the same another flag by them. Since precisely four flags are associated with each edge, this implies the following inequality, as well as Weinberg’s inequality: Sym(G) ≤ 4 E(G) The 2-cell embedding of G or G itself is said to be full-symmetric if G attains this upper bounds ( Sym(G) = 4 E(G) ). It is obvious that G is full-symmetric if and only if so is its dual G∗ since they have the same set of flags. In this terminology, Weinberg’s theorem states that a 3-connected graph embedded on the sphere is full-symmetric if and only if it is isomorphic to one of platonic graphs. Our purpose is to classify ∗ Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-Ku, Yokohama 223, Japan. Email: [email protected] † Department of Mathematics, Faculty of Education, Yokohama National University, 156 Tokiwadai, Hodogaya-Ku, Yokohama 240, Japan. Email: [email protected]

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such full-symmetric embeddings on other closed surfaces. Two embeddings fi : Gi → F 2 of graphs Gi (i = 1, 2) are said to be equivalent to each other if there exist an isomorphism σ : G1 → G2 and an auto-homeomorphism h : F 2 → F 2 with hf1 = f2 σ. THEOREM 1. Every full-symmetric embeddings on the projective plane is equivalent to one of the following: (i) the unique embedding of K6 , (ii) the unique embedding of the petersen graph, (iii) the unique embedding of K4 with three quadrilateral faces, (iv) its dual embedding with three vertices of degree 4. (v) a cycle embedded along an essential closed curve, and (vi) its dual, which is a wedge of loops. Remark that the embeddings of K6 and of the petersen graph in the projective plane are dual to each other, as well as the other two pairs in the theorem. THEOREM 2. Every full-symmetric embeddings on the torus is equivalent to one of the following embeddings: (i)

T (p, 0, p), T (3r, r, r)

(ii)

Q(p, 0, p), Q(2r, r, r)

(iii)

T ∗ (p, 0, p), T ∗ (3r, r, r)

The notation T (p, q, r) stands for a standard form of 6-regular triangulations on the torus with dual T ∗ (p, q, r) while Q(p, q, r) is a 4-regular quadrangulation and is self-dual. (A graph G embedded on a closed surface F 2 is said to be self-dual in general if there is a homeomorphism h : F 2 → F 2 such that h(G) = G∗ and h(G∗ ) = G for the dual G∗ of G.) The former has been already classified in [1] and in [4], and both will be described with the proof of the theorem in Section 3. Their classification and identified structures lead us to more concrete arguments. If graphs should be simple, then we need conditions p ≥ 3 and r ≥ 2. Otherwise, p and r are just positive integers. Similarly, the classifications of 6-regular triangulations and of 4-regular quadrangulations of the Klein bottle work essentially to prove the next theorem. The former can be found in [6] while the latter will be given in Section 4. THEOREM 3. There is no full-symmetric embedding of any graph on the Klein bottle. Our proof of the above theorem will lead us to a conjecture that every nonorientable closed surface, except the projective plane, admits no full-symmetric embedding. On the other hand, we can construct at least two full-symmetric embedding on every orientable closed surface and show that there are not so many those, by easy combinatorics. THEOREM 4. Every hyperbolic closed surface, orientable or nonorientable, admits at most a finite number of full-symmetric embeddings.

2. General observations

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In this section, we shall obverse some general facts on full-symmetric embeddings and prove Theorems 1 and 4 from those. LEMMA 5. Let G be a graph 2-cell embedded on a closed surface F 2 . If G is full-symmetric, then both G and its dual G∗ are regular graphs. Proof. Let u and v be two vertices of G. Choose two flags of edges with u and v one of their ends, respectively. If G is full-symmetric, then there exists an auto-homeomorphism of F 2 which carries one of these flags onto the other. The restriction of such a homeomorphism to G is an automorphism of G which maps u to v. Thus, u and v have the same degree and hence G is regular. Similarly, there is an auto-homeomorphism of F 2 for any pair of faces of G which carries one to the other. The regularity of G∗ follows from this. A graph 2-cell embedded on a closed surface which and whose dual are regular is often called a regular map and has been discussed in many frameworks. So the following lemma might be found in some papers. LEMMA 6. Let G be a r-regular graph 2-cell embedded on a closed surface F 2 with dual G∗ r∗ -regular. If the Euler characteristic χ(F 2 ) of F 2 is not 0, then we have: V (G) =

V (G∗ ) =

−2χ(F 2 )r∗ , rr∗ − 2r − 2r∗

−2χ(F 2 )r rr∗ − 2r − 2r∗

Proof. Let V and V ∗ denote the number of vertices of G and of G∗ , respectively, and E the number of edges of G. By Euler’s formula and Handshaking Lemma, V − E + V ∗ = χ(F 2 ),

2E = rV,

2E = r∗ V ∗

Substituting the last two into the first, we have the following simultaneous equation of matrix form: ( )( ) ( ) 2−r 2 V 2χ(F 2 ) = 2 2 − r∗ V∗ 2χ(F 2 ) Solve this. Since V (G) is a positive number, if χ(F 2 ) > 0 (or < 0), then rr∗ − 2r − 2r∗ < 0 (or > 0). On the other hand, if χ(F 2 ) = 0, then the simultaneous equation in the above proof has to be indefinite and hence rr∗ − 2r − 2r∗ = 0, which is the determinant of its coefficient matrix. Table 1 shows the possible pairs of values of degrees r and r∗ with assumption r ≤ r∗ . Note that rr∗ − 2r − 2r∗ 0 is equivalent to the following inequality: π π π + + ∗ >, =, < π 2 r r This is the condition for the three angles π2 , plane or the hyperbolic plane, respectively.

π π r , r∗

of a right triangle in the unit sphere, the Euclidean

Proof of Theorem 1. From the formula in Lemma 6 and Table 1, we get Table 2 for the possible numbers of elements of G and G∗ . It is easy to see that the only embeddings listed in the theorem realize the cases given in Table 2. Theorem 4 is an immediate consequence from Lemma 5 and the following theorem. THEOREM 7. Every hyperbolic closed surface admits only finitely many graphs which and whose dual are regular.

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Surfaces elliptic χ>0

parabolic χ=0 hyperbolic χ (λ − 2)p This implies that λ = 1 or 2, that is, (a) r = p − 2q, or (b) r = 2p − 2q. Under our second assumption, it is obvious that if p = r, then q = 0 since (r, p) projects to uq and (p, p) to v = u0 . This is the first condition (i) in the lemma. So we may assume that r < p hereafter. 6

p t

q - q t × t

t 0

p

2p

−r Figure 1: T (p, q, 2p − 2q) Figure 1 illustrates Case (b). The vertical line through (2p − 2q, 0) = (r, 0) is one of the lifts of C0 and (2p − 2q, p − q) projects to v. On the other hand, the rotation ρ deforms the fundamental region Ωp,q,r into the parallelogram determined by two vectors (p, p) and (0, −r) based at the origin (0, 0), which should be another fundamental region. However, (2p − 2q, p − q) is contained in the interior of the second fundamental region, which contradicts that this point is equivalent to (0, 0), corresponding to v. Thus, this is not the case. 6

6 p t

t q

t

t t 0

r

p

−r

Figure 2: T (p, q, p − 2q) Figure 2 describes the situation of Case (a) with q ̸= 0. (If q = 0, then we have T (p, 0, p).) Let Ω′ be the second fundamental region given in the previous paragraph. On its boundary, (0, 0), (p, p) and (p − q, q) project to v and hence the vertical line through the third point has to be one of the lifts of C0 . Since one through (p − 2q, 0) also covers C0 , the distance between these two lines, equal to q, has to be devided by r. If r < q, then (p − 2q + r, p − 2q) would be contained in Ω′ and project to v, contrary to Ω′ being a fundamental region. Thus, r = q and we have p = 3r. This is the second type (ii) in the lemma. Conversely, it will be clear, from these observations, that T (p, 0, p) and T (3r, r, r) are full-symmetric.

Following the arguments for 6-regular triangulations in [4] with suitable modification, we can obtain the standard forms of 4-regular quadrangulations Q(p, q, r) and thier classification. We shall give only their description without detailed arguments. (Every 4-regular quadrangulation on the torus has a structure call “a cycle-bundle over a cycle”. Such structures have been classified completely in [6].) Let G be a 4-regular quadrangulation of the torus, that is, an embedding of 4-regular graph with quadrilateral faces. Let v be any vertex with neighbors v0 , v1 , v2 , v3 lying around v in this cyclic order. Similarly to the case of T (p, q, r), consider a geodesic cycle C0 = u0 u1 · · · up−1 through vv0 = u0 u1 and a geodesic path w0 w1 · · · wr through vv1 = w0 w1 with wr = uq . Then G is denoted by Q(p, q, r). ˜ in the universal covering space of the torus, as described in the Erase all the slope 1 lines from G previous. Then the infinite quadrangulation of the Euclidean plane, consisting of vertical and horizontal lines, will be obtained and G will be given as its quotient by the same group action generated by τm,n . The same idea as for T (p, q, r) works for Q(p, q, r), but more simply. LEMMA 9. A 4-regular quadangulation Q(p, q, r) of the torus is full-symmetric if and only if either (i)

p = r and q = 0, or

(ii)

p = 2r and q = r.

Proof. Similarly to the proof of Lemma 8, consider the extension of the local reflexion τ with axis v0 vv2 and the rotation ρ around v with ρ(vi ) = vi+1 . First suppose that τ extends. Then (−r, p + q) and (−r, p − q) should be equivalent to (0, 0) and project to v. Thus, p + q ≡ p − q and hence 2q ≡ 0 mod p. Since q < p, this implies either q = 0 or 2q = p. 7

Moreover suppose that ρ extends. Then (p, 0) should be equivalent to (0, 0). If q = 0, then (r, 0) projects to v and has to coincide with (p, 0) since the interval between (0, 0) and (p, 0) cannot contain any inner point which projects to v. Thus, we have Q(p, 0, p) in this case. If 2q = p, then (2r, 0) should project to v and hence 2r = p. This is the second type Q(2r, r, r) in the lemma.

Proof of Theorem 2. As is shown in the previous section, the possible pairs of degrees of a full-symmetric graph embedded on the torus and its dual are (3, 6), (4, 4), (6, 3). The third types are realized as 6-regular triangulations on the torus and has been classified in Lemma 8; namely T (p, 0, p) and T (3r, r, r). On the other hand, Lemma 9 has classified the second types, which are Q(p, 0, p) and Q(2r, r, r). The first types are the duals of the third ones. So they are T ∗ (p, 0, p) and T ∗ (3r, r, r).

4. Klein bottlal case If there were a full-symmetric embedding of a graph G on the Klein bottle, then G and its dual G∗ would be regular with degrees (3, 6), (4, 4) or (6, 3), as well as in the toroidal case. So we shall show the classification of those graphs on the Klein bottle and conclude the lack of full-symmetric embeddings through this section by observing their structures. Negami has already classified the 6-regular triangulations of the Klein bottle, corresponding to (6, 3), in his thesis [6] and in [5] as follows. Let Hp,r+1 be the graph obtained from T (p, 0, r) by cutting along a geodesic cycle C0 of length p. Then Hp,r+1 contains r + 1 disjoint cycles C0 , C1 , . . . , Cr placed in parallel. Let u0 u1 · · · up−1 and u′0 u′1 · · · u′p−1 be the sequence of vertices along C0 and Cr , respectively, so that ui and u′i are identified in T (p, 0, r). The handle type Kh(p, r) can be obtained from Hp,r+1 by identifying ui and u′−i with indices taken modulo p. Clearly, Kh(p, r) can be embedded on the Klein bottle to be a 6-regular triangulation. The union of r cycles C0 = Cr , C1 , . . . , Cr−1 covers all the vertices of Kh(p, r) and is called the geodesic 2-factor of Kh(p, r). It is an important fact that any other geodesic walk, except C0 , . . . , Cr−1 , cannot be a geodesic cycle. The crosscap type Kc(p, r) is constructed in the following two ways, depending on the parity of p. When p = 2d is even, prepare Hp,r+1 and identify each pair of ui and ui+d on C0 and of u′i and u′i+d on Cr . Then the resulting graph Kc(p, r) can be embedded on the Klein bottle as a 6-regular triangulation so that C0 and Cr are identified to be two cycles of length d, say C¯0 and C¯r , lying along the center lines of two disjoint Möbius bands, or crosscaps. The union of C¯0 , C1 , . . . , Cr−1 , C¯r is the geodesic 2-factor of Kh(p, r) with p even. When p = 2d + 1 is odd, prepare Hp,r and add new edges ui ui+d and u′i u′i+d for i ≡ 0, 1, . . . , p − 1 (mod p) to complete Kc(p, r). This Kc(p, r) also can be embedded on the Klein bottle as a 6-regular triangulation so that C0 and Cr−1 bound two Möbius bands triangulated by the added edges. The geodesic 2-factor of Kc(p, r) consists of r cycles C0 , C1 , . . . , Cr−1 of the same length p. By Negami’s classification, any 6-regular triangulation on the Klein bottle is equivalent to either Kh(p, r) or Kc(p, r) with suitable parameters p and r and it has a unique geodesic 2-factor, that is, a disjoint union of geodesic cycles which covers all its vertices. The uniqueness of the geodesic 2-factor in Kh(p, r) and Kc(p, r) implies that they are not full-symmetric since there is no auto-homeomorphism of the Klein bottle which realizes an automorphism of such a graph carrying an edge on the geodesic 2-factor to one not on it. Thus, we have concluded that there exists neither a full-symmetric embedding on the Klein bottle corresponding to (6, 3) nor to (3, 6). Now we shall classify the 4-regular quadrangulations on the Klein bottle, showing three kinds of their standard forms as follows. Prepare the Cartesian product Pp+1 × Pr+1 of two paths naturally embedded in a rectangle of size p × r, say Rp,r , where Pn stands for the path with n vertices. Identify the pair of horizontal sides of length r in parallel and the pair of vertical sides of length p in anti-parallel to get the Klein bottle. The resultng 4-regular quadrangulation in the Klein bottle is called the grid type and is denoted by Qg (p, r). The simple closed curve which comes from the vertical sides of Rp,r cuts open the Klein bottle into an annulus. Such a simple closed curve on the Klein bottle is called a meridian in general. On the other hand,

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Figure 3: The grid Rp,r the horizontal sides of Rp,r form a simple closed curve whose tubular neighborhood is homeomorphic to the Möbius band, called a longitude. There is another longitude joining the middle points of the vertical sides. If p is even, then the second longitude also can be obtained as a geodesic cycle of length r in Qg (p, r). Now prepare R2p,r and identify the pair of its horizontal sides to get an annulus with cycles C and C ′ of length p on its ends. Attach two Möbius bands to C and C ′ and add p edges to join antipodal pairs of vertices on C in one of the two Möbius bands and those to the other. The resultng 4-regular quadrangulation in the Klein bottle is called the ladder type and is denoted by Ql (2p, r). This quadrangulation Ql (2p, r) has what are called “Möbius ladders” at both ends and contains no longitude given as its geodesic cycle. Note that Ql (2p, r) contains p meridians of length 2r + 2 each of which is obtained from two horizontal paths in R2p,r and two edges in the two Möbius ladders. The third type of 4-regular quadrangulation, called the mesh type, is constructed as follows. Consider the rectangle Ω = {(x, y) ∈ R2 : 0 ≤ x ≤ r, 0 ≤ y ≤ 2p} and the set of points V = {(x, y) ∈ Ω ∩ Z 2 : x − y ≡ 1 (mod 2)}. Let Mp,r be the graph with vertex set V such that each vertex (x, y) is adjacent to (x ± 1, y ± 1) ∈ V . Then the union of its edges forms a mesh dividing Ω with lines of slopes ±1. Identify the pair of horizontal sides of Ω in parallel to get a cylinder and next glue its two ends incoherently to unify the vertices along them in pair. The resulting graph in the Klein bottle is the mesh type and is denote by Qm (p, r). Note that Qm (p, r) can be determined uniquely up to equivalence, not depending on the way to glue the ends of the cylinder. u u u u @ @ @ u @u @u @u @ @ @ @ @u @u @u @u @ @ @ u @u @u @u @ @ @ @ @u @u @u @u @ @ @ u @u @u @u @ @ @ @ @u @u @u @u Figure 4: The mesh Mp,r

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THEOREM 10. Every 4-regualr quadrangulation on the Klein bottle is equivalent to one of Qg (p, r), Ql (2p, r) and Qm (p, r) with suitable parameters p, r. Proof. Consider the double covering π : T 2 → K 2 from the torus T 2 onto the Klein bottle K 2 with covering transformation τ : T 2 → T 2 . That is, τ is the orientation-reversing auto-homeomorphism over T 2 of period 2 such that π(x) = π(y) (x ̸= y ∈ T 2 ) if and only if y = τ (x). Let G be a 4-regular ˜ = π −1 (G) the pre-image of G via π. Since the projection quadrangulation on the Klein bottle K 2 and G ˜ π is a local-homeomorphism, G also is a 4-regular quadrangulation on the torus T 2 . ˜ should be Q(p, q, r) for suitable parameters p, q, r. To read these We have already known that G parameters, we take a geodesic cycle C0 of length p starting at a vertex v and find r disjoint cycles C0 , C1 , . . . , Cr−1 of the same length which are placed in parallel over the torus and cover all the vertices ˜ The union of these cycles is a geodesic 2-factor of G, ˜ say H. We can choose another geodesic cycle of G. C0′ through v and orthogonal to C0 and find the second geodesic 2-factor H ′ = C0′ ∪ · · · ∪ Cr′ ′ −1 . Since ˜ we have two possibilities: τ (H) has to be a geodesic 2-factor of G, (i) τ (H) = H, τ (H ′ ) = H ′ . (ii) τ (H) = H ′ , τ (H ′ ) = H. First suppose Case (i). Since τ reverses the orientation over the torus, the rotation around τ (v) induced by τ is not compatible to that around v. It follows that τ reverses the directions of cycles Ci in H and preserves those in H ′ , without loss of generality. If τ (Ci ) = Ci for some i, then Ci would contain two fixed points of τ , contrary to τ being the covering transformation. Thus, τ (Ci ) ̸= Ci for each i. This implies that r is an even number, say 2s, and that τ (Ci ) = Ci+s with indices taken modulo r. ˜ is Let A be one of the two annuli bounded by C0 ∪ Cs which contains C0 , C1 , . . . , Cs−1 . Then A ∩ G isomorphic to the Cartesian product of a cycle of length p with a path of length s and π(A) covers the ˜ by identifying C0 and whole of the Klein bottle with π(C0 ) = π(Cs ). So G can be obtained from A ∩ G Cs . Let u0 , . . . , up and v0 , . . . , vp be vertices lying along C0 and Cs in this order, respectively, so that ui ˜ The identification of C0 with Cs may be given by either and vi are joined by a geodesic path in A ∩ G. ui = v−i or ui = vp−1−i . In the first case, cut the annulus A into a rectangle along the geodesic path joining u0 and v0 . Then the grid Rp,s will be obtained. This implies that G is equivalent to Qg (p, s). If the second case cannot be reduced to the first, then p is an even number 2t and G is equivalent to Ql (2s, t). The two geodesic paths joining u0 to v0 and up−1 to vp−1 form the “rim” of one of the two Möbius ladders in Ql (2s, t) and those joining ut−1 to vt−1 and ut to vt forms the other. ˜ Then R(G) ˜ is a 4-regular quadrangulation Now suppose Case (ii) and consider the radial graph R(G). ˜ Let A′ be the annulus which covers on the torus which τ leaves invariant, and Case (i) happens for R(G). the Klein bottle corresponding to the above A and let D0 , D1 , . . . D2k be the cycles of length 2h in A′ , ˜ corresponding to C0 , C1 , . . . , Cs , such that D0 ∪ Dk bounds A′ . Cut A′ along the geodesic path in R(G) ˜ ∗ and add the edges of G ˜ after removing the vertices of G ˜ ∗ . Then, starting from a vertex belonging to G Mh,k will be obtained in the resulting rectangle. This implies that G is equivalent to Qm (h, k). Proof of Theorem 3. As we have already shown, there is no full-symmetric embedding of degrees (6, 3) and (3, 6). To show the lack of those of (4, 4), it suffices to observe that any type of a 4-regular quadrangulation of the Klein bottle is not full-symmetric. The grid type and the ladder type include edges lying on a meridian given as a geodesic cycle and ones not lying on such a cycle. These edges cannot be transfered to each other by any element in their symmetry groups. In the mesh type, any face cannot be rotated through 90◦ . For if it could be, then a meridian would be transfered to a simple closed curve which is not a meridian. Thus, each type of 4-regular quadrangulations is not full-symmetric.

References [1] A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217. 10

[2] D.S. Archdeacon and R.B. Richter, The construction and classification of self-dual spherical polyhedra, J. Combinatorial Theory, Ser. B . 54 (1992), 37–63. [3] D. Archdeacon and S. Negami, The construction of self-dual projective polyhedra, J. Combinatorial Theory, Ser. B . 59 (1993), 122–131. [4] S. Negami, Uniqueness and faithfulness of embedding of toroidal graphs, Discrete Math. 44 (1983), 161-180. [5] S. Negami, Classification of 6-regular Klein-bottlal graphs, Res. Rep. Inf. Sci. T.I.T. A-96 (1984). [6] S. Negami, Uniqueness and faithfulness of embedding of graphs into surfaces, Doctor thesis, Tokyo Institute of Technology, 1985. [7] L. Weinberg, On the maximum order of the automorphism group of a planar triply connected graph, SIAM J. Appl. Math. 14 (1966), 729-738.

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