imbeddings are into orientable surfaces for n = 1 or 2 (mod 3) and into orientable .... the non-orientable surface N15 (see [22]); another comes from imbedding a.
EFFICIENT IMBEDDINGS OF FINITE PROJECTIVE PLANES
ARTHUR T. WHITE [Received 2 February 1993—Revised 15 November 1993]
ABSTRACT We use index-one voltage graph imbeddings of Cayley graphs (Cayley maps) for the groups Zn2+n + v where n is a prime power, to depict the finite projective planes PG(2, n). Subject to the condition that Z n 2 + n + i act regularly not only on the images of the points and of the lines of PG(2, n), but also on each orbit of the region set for the imbedding, the imbeddings constructed have maximum efficiency, as they attain the upper bound for the euler characteristic of an ambient space. The imbeddings are into orientable surfaces for n = 1 or 2 (mod 3) and into orientable pseudosurfaces for n = 0 (mod 3). An easy modification of the imbeddings for n = 2 (mod 3) gives the genus of the (n + l,6)-cages for these values. Efficient imbeddings are constructed also for the 'doubled' designs arising from the pairing of each PG(2, n) with its dual. The map automorphism groups are studied for all these imbeddings (A = 1 and 2). Deletions from the imbeddings produce asymptotically efficient imbeddings for the affine planes AG(2, n).
1. Introduction A finite projective plane of order n (n ss 2) consists of n2 + n + 1 points, each belonging to n + 1 lines, and n2 + n + 1 lines, each consisting of n + 1 points. Each pair of distinct points lie on a unique line, and each pair of distinct lines meet in a unique point. Thus the plane and its dual each determine a symmetric (n2 + n +l,n + 1,1) balanced incomplete block design. It will often be notationally convenient to equate a plane with its design. The classical desarguesian planes PG(2, n) exist precisely when n is a prime power. No projective plane of any other order is yet known to exist; many non-existence results have been obtained. (See, for example, [15] or [23]. For much more information about projective planes, see also [9] or [17].) The planes PG(2, n) may not be the only planes of order n; for example, PG(2, n) is the only plane of order n, for n «=8, but there are three planes of order 9 in addition to PG(2,9) (see [3], for example). Moreover, if n = pm s* 9, with m^2, there is always a nondesarguesian plane of order n [9, p. 144]. In this paper, we consider only the planes PG(2, n). The smallest projective plane, the Fano plane PG(2,2), is usually depicted as
The author is indebted to Western Michigan University for funding his 1991-92 sabbatical leave, and to the Mathematical Institute and Wolfson College, University of Oxford, for receiving him as a visitor. 1991 Mathematics Subject Classification: primary 05C10, 51E15; secondary 05C25, O5BO5, 05B10,
05B25. Proc. London Math. Soc. (3) 70 (1995) 33-55.
34
ARTHUR T. WHITE
0 1 2 3 4 5 6
1 2 3 4 5 6 0
3 4 5 6 0 1 2
0 6 5 4 3 2 1 FIG.
6 5 4 3 2 1 0
4 3 2 1 0 6 5
1
in Fig. 1. The corresponding (7,3, l)-balanced incomplete block design (BIBD) is a Steiner triple system, and the blocks are listed below the drawing as design D. The isomorphic design -D is obtained by multiplying each object of D by - 1 or, equivalently, by taking additive inverses in Z7. Together the two Steiner triple systems form one (7,14,6,3,2)-BIBD, a 2-fold triple system. The representation of PG(2,2) given by Fig. 1 has several serious deficiencies: (i) the line {1,2,4} has a different shape to that of the other lines; (ii) each other line has two 'end' points and one 'middle' point; yet there is no notion of 'betweenness' in PG(2,2); (iii) there are three 'crossings' in the figure, that have no meaning in the geometry; (iv) one cannot tell the points 0, 1,2, and 4 are each on three lines, by looking at small neighbourhoods of those points. Less serious problems with Fig. 1 are: (v) the regular action of Z7 on the points and on the lines of PG(2,2) is not visually apparent; (vi) the relationship of PG(2,2) with the inverse plane, -PG(2,2), is obscured. Existing depictions of PG(2, n) for n > 2 seem even more inappropriate. Our goal in this paper is to depict the planes PG(2, n) with more accuracy, in a locally 2-dimensional way, and to do this as efficiently as possible. In § 3 we overcome all but objection (vi) above, and in § 5 we address all six objections. In the remainder of this section we discuss the contributions of others towards our goal, give relevant definitions, and summarize our results. In § 2 we provide the necessary background about PG(2, n) and its related difference sets, and about the voltage graph constructions we will use, illustrating how these ideas fit together with the example of PG(2,5). Section 4 is devoted to the cages arising from PG(2,«), and §6 to the affine planes AG(2,«). Heffter [16] was the first to study the connection between block designs and graph imbeddings. Later work has been by Emch [10], Alpert [1], White [30],
EFFICIENT IMBEDDINGS OF FINITE PROJECT1VE PLANES
35
Anderson and White [4], Anderson [2], Jungerman, Stahl, and White [18], and Rahn [20,21]. The imbeddings are into surfaces (closed 2-manifolds, either orientable or non-orientable), into pseudosurfaces (resulting from making finitely many identifications, of finitely many points each, on a surface), or into generalized pseudosurfaces (resulting from making finitely many identifications, of finitely many {joints each, on a topological space of finitely many components, each of which is homeomorphic to a pseudosurface, with the final topological space being connected). In the latter two cases, we require that each singular point be the image of a vertex, in any graph imbedding. The central correspondence is between 2-fold triple systems of order m (BIBDs with block size k = 3 and A = 2; the case m = 7 occurs in Fig. 1) and triangular imbeddings of complete graphs of order m into generalized pseudosurfaces. The idea behind the correspondence is that any two distinct objects in the design are modelled by distinct vertices of Km, and thus form an edge. This edge bounds two triangular regions of the imbedding, and these are the A = 2 blocks of size k = 3 the two objects belong to. If, in addition, the dual of the imbedding is bichromatic, then selecting the triangles of either colour yields a Steiner triple system. Very few graph imbeddings have been found heretofore giving block designs having k > 3. This paper will provide infinitely many. One advantage of this topological point of view for block designs is expressed in Theorem 7 of [30]: if two 2-fold triple systems of order m determine topological spaces that are not homeomorphic, then the designs are not isomorphic. For example, one (13,52,12,3,2)-BIBD arises from a triangular imbedding of Kn on the non-orientable surface N15 (see [22]); another comes from imbedding a 6-regular subgraph of /C13 = GA(Zn) on one torus, the complementary subgraph on a second torus (see [4]), and then identifying the two vertices labelled /, with 0=s/*£l2. The result is the generalized pseudosurface (25], 13(2)), having characteristic -13, clearly not homeomorphic to N]5 (necessarily also of characteristic — 13). Thus the designs are,different. Walsh [28] introduced the incidence graph G(D) associated with design D as follows. The vertices of G(D) are the objects and the blocks of D, and object x is adjacent to block y in G{D) precisely when x e y in D. Thus G(D) is a bipartite graph. The ideas we have been discussing are illustrated in Fig. 2, for D = PG(2,2) and its related structures. In Fig. 2(a) we see a triangular imbedding of the Cayley graph K7 for the group Z7, in the torus and having bichromatic dual. (The unshaded regions give design D of Fig. 1; the shaded regions give —D.) In Fig. 2(b) we show how to modify Fig. 2(a) to obtain an imbedding of G(D), called the Heawood graph for this case (n =2), also in the torus. (The process is seen to be reversible.) In Fig. 2(c) we show that Fig. 2(a) is a 7-fold covering of a much simpler toroidal imbedding, and this foreshadows the voltage graph constructions we will be employing in §§ 3 and 5. Walsh, having found a copy of K33 in G(D) for £> = PG(2,2), utilized his rendering of Fig. 2(b) in [28] to establish that PG(2,2) has genus 1 and then raised the problem of determining the genus of PG(2, n) in general. This paper addresses that problem. Singerman [25] also considered the problem of imbedding the planes PG(2, n) in (Riemann) surfaces, but in the context of the map automorphism group acting regularly on the bits (x, y) of G(D), where point x lies on line y of PG(2, n) = D. He claimed that there are exactly four such imbeddings: for n = 2 on the torus
36
ARTHUR T. WHITE
(C) FIG.
2
and on the triple torus (Klein's Riemann surface) and for n = 8 on the orientable surfaces of genera 220 and 252. In [12] Fink and White find a fifth: for n = 8 on the orientable surface of genus 147. In [11] Fink showed that no sharply flag-transitive plane (desarguesian or otherwise) of order n =* 3600 exists, except for n = 2 or 8. Thus the regular imbeddings of Singerman, Fink, and White cover all cases, for n =s 3600. Our development in § 3 will overlap with that of [25] and [12] for the first and last of the five imbeddings, and in § 5 for the first and third. The definition of the genus parameter for block designs is reminiscent of that for the genus parameter for groups (see [29,32]). The unifying theme is that of depicting an abstract mathematical object graphically, in a locally 2-dimensional setting, as efficiently as possible. Let Sk denote the orientable surface of genus k (k 2=0). The genus y(G) of a graph G is a smallest k such that G imbeds in Sk. The genus y(T) of a group T is then the minimum, taken over all generating sets A for T, of the values y(G±(T)), where the Cayley graph GA(F) has vertex set T and edge set {{g,g8}\ g e T, 8 G A U A " 1 } , where A"1 = {5 -1 | 8 E A}. Now let D be a block design. The genus y(D) of D (see [18] and [21]) is defined by y(D) = y(G(D)). But this may be too restrictive for our purposes, as the Heffter/Alpert correspondence requires more than just orientable surfaces (even for 2-fold triple systems). We denote the non-orientable surfaces of genus
EFFICIENT IMBEDDINGS OF FINITE PROJECTIVE PLANES
37
h (h > 0) by Nh, and let 5, 5', and S" denote a generic surface, pseudosurface, and generalized pseudosurface respectively. Then the characteristic x(D) of the design D is the maximum euler characteristic among all surfaces S in which G(D) imbeds. (Note that *(S) = 2-2A: if S = Sk and x(S) = 2-h if S = Nh.) The pseudocharacteristic x'(D) of D is defined to be the maximum euler characteristic among all pseudosurfaces 5' in which G(D) can be imbedded. (See [19] or [31].) The generalized pseudocharacteristic x"(D) is then defined analogously for S" (see [31]), but will not be needed here. The group Zni+n+1 acts regularly (that is, as a sharply 1-transitive permutation group, with order equal to degree) on both the points and lines of PG(2,n). Ideally, we would like this feature to extend to the regions of an imbedding for PG(2, n), that is, an imbedding of PG(2, n) as the modification of an imbedding of G(PG(2,n)) as indicated in Fig. 2(b). Specifically within each region of the imbedding of G(D) (D = PG(2,n)) we add an edge between successive point vertices in the region boundary, and then delete all edges and line vertices of G(D). 1.1. DEFINITION. The regular pseudocharacteristic of PG(2, n), x'r(PG(2, n)), is the maximum x s u c n that G(PG(2, n)) imbeds in a pseudosurface S', of characteristic x'> with Zn2+n+i acting regularly (as a group of map automorphisms) on each orbit of the region set for the modified imbedding. Our primary goal is to determine £r'(PG(2, n)) for every prime power n. A secondary goal is to maximize the euler characteristic among all generalized pseudosurfaces in which G(PG(2, n)) imbeds. As adding handles, adding crosscaps, or making vertex identifications all lower the euler characteristic of the ambient space, maximum characteristic means maximum efficiency of the imbedding. Having maximized the characteristic, we then want a generalized pseudosurface which is in fact a pseudosurface to receive our depiction of PG(2, n). We will always achieve.this. Next, we want the ambient space to be orientable, and in the regular case we always achieve this, too. Finally, we want the ambient space to be a surface; this is not always possible. We give an example to clarify the distinction among the various parameters. We can construct three imbeddings, which we denote by Iu I2, and 73, each depicting PG(2, 4) as a modification of an imbedding of G(PG(2, 4)). (i) The imbedding /j is into the orientable surface S22, of characteristic -42, and Z21 produces two region orbits, each of length 21, yielding ^/(PG(2,4)) = -42. (ii) The second, /2, is into the non-orientable surface NM, of characteristic -32, and Z2i produces two region orbits of length 21, one of length 7, and one of length 3. Thus Z21 acts as a group of map automorphisms, but not regularly on each region orbit. (iii) The third imbedding 73 is into N30, of characteristic -28, and Z21 does not act on the set of regions. The action is by the subgroup ) is called a voltage graph. Let {K, Y,) be 2-cell imbedded in the orientable surface S. Then there exists a 2-cell imbedding of Kx^T into an orientable surface S and a (possibly branched) covering projection p: S —>S such that (i) p-l(K)
= KxT;
(ii) ifb is a branch point of multiplicity m, then b is in the interior of a region R such that \R\^ = m; (iii) if R is a k-gonal region for K in S, then p~\R) has |r|/|/?|^ components, each a k \R\4-g0nal region for Kx^T in S.
40
ARTHUR T. WHITE
If |/?|^ = 1, we say that R satisfies the Kirchoff Voltage Law (KVL). If the KVL holds for every R determined by K in S, then we say that imbedding satisfies the KVL. Let p , q, and r denote the number of vertices, edges, and regions for K in 5, and similarly p , q, and f for K = K X 0 F in S. We always have p = \T\p and q = |F| q\ if the KVL holds globally (i.e. for the imbedding) then (by Proposition 2.2(iii)) we also have f = |F| r. It follows immediately (see Theorem 2.3(ii)(a)) that the euler characteristics % and x of S and 5 respectively are related by % = 1^1 XFinally, we say that the imbedding (either above or below) is of index i (i G N) if | V(AT)| = /. Now we are ready for the special case of the voltage graph theory we need. For an index-one imbedding of a voltage graph (K, F, ), each pair e, e"1 from K* corresponds to a loop of K which carries elements (e) and ((f>(e))~x of F. Suppose that these pairs are pairwise disjoint, and that (e) is never the identity (e). Form A g F - j e } by selecting exactly one element from each pair {(e), {He))-')2.3. THEOREM. Let (K, F, ) be an index-one voltage graph, 2-cell imbedded in an orientable surface S and satisfying the KVL. Let A, formed as above, generate F. (i) The graph KxT = G^(T). (ii) The imbedding of GA(T) in S satisfies: (b) let R be bounded by (eu e2,..., em) and let the closed walk (g, g{e\), g{e\)4>(ei)> -, g4>(e,)(f>(e2) - 4>(em-x)) bound a polygonal disk Rg; then p~'l(R) = {Rs\ g e F}; (c) F is isomorphic to a group of map automorphisms acting regularly on each orbit (p~\R)) of the region set. = V(GA(T)) be given by d(u,g) = Proofs, (i) Let 0: V(Kx4>T) = {u}xT^T g. Then 6 is a graph isomorphism. (ii)(a) We always have p = \T\p and q = |F| q. By Proposition 2.2(iii), using the KVL, we now also have f = |F| r. Thus ^(5) = p - q + f = \T\ (p - q + r) = (ii)(b) This follows immediately, from the definition of edges in A^X^F, part (i) of this theorem, the KVL, and Proposition 2.2(iii). (ii)(c) Always F ^ Aut GA(F), in its left regular representation, that is, F = {6h\ h G F}, under composition, where 6h: F—>F is given by 6h{g) = hg. We claim that 9h preserves oriented region boundaries, so that (with respect to isomorphism) F *£ Aut M, where M is the map consisting of GA(F) imbedded in 5; and that the region orbits of the action of F are precisely the sets p~\R). Since \p~\R)\ = |F| for each R, it would follow that F acts regularly on each orbit. To verify the first claim, we need only note that 6h(Rg) = Rhg e p~\R), if Rg E p~\R). To verify the second claim, let Rgu Rg2 G p~\R). Then 6g3gl-i(Rgi) = Rg2, so that F is transitive on p~\R). But no orbit can be larger than the group acting on the set containing the orbit, so p" 1 (R) is an orbit of this action. An easy modification of Theorem 2.3 shows that if 5 is a pseudosurface, then S
EFFICIENT IMBEDDINGS OF FINITE PROJECTIVE PLANES
41
will be also, with |F| singular points. (Each point is the image of a vertex of GA(F) covering the one singular point below, namely the image of the one vertex of K. The modification is obtained by reversing the identification on S, applying Proposition 2.2, then making appropriate identifications on S.) For a nonorientable analogue of Proposition 2.2, leading to a non-orientable analogue of Theorem 2.3, see, for example, Chapter 11 of [31]. We illustrate the ideas of this section with the example of PG(2, 5). We start with GF(5) = {0,1,2,3,4}, and f(x) = x2 - x - 2, which is clearly monic and cubic over GF(5). As f(y)^0, for all y e GF(5), f(x) is also irreducible. Thus we set x3 = x + 2, and calculate increasing powers of x: xn = = l, x, x 3*+2, ..., *31 = 2eGF(5). Thus = Z124 = GF(53) - {0} and x is primitive. Moreover, we take {x'}o° as coset representatives for (GF(53) - {0})/(GF(5) - {0}) and Z 3] for the points of PG(2,5) (multiplying powers of x corresponds to adding exponents.) The line Lo = {0,1, 3, 8,12,18} is readily checked to be a planar difference set, and we calculate ; = 6~ J (-(0 + 1 + 3 + 8 + 12 + 18)) = ( - 5 ) ( - l l ) = 24 in Z31. Thus L = L M = {24, 25, 27,1, 5, 11}; we check that 5L = L. Reorder L to {1,11,5,24,25,27} and use successive differences to form A = {10, - 6 , -12,1,2,5} for F = Z31. We note that (as if by magic, but the 'magic' will be explained in § 3): (1) 1 0 - 6 - 1 2 + 1 + 2 + 5 = 0, (2) 10-12 + 2 = 0, (3) - 6 + 1 + 5 = 0; these three equations give the KVL property for the voltage graph imbedding of Fig. 3.
There are six edges in K, corresponding to the six elements in A, and three regions. There is only one vertex (after identification of the two occurrences of each edge, matching up the arrows), as seen by the (clockwise) ordering of the generators and their inverses: (-2,10, - 5 , - 6 , -10, -12,6,1,12,2, -1,5). The surface S in which K is imbedded is orientable, as each edge appears once in each direction, among all the (clockwise, say) region boundaries. The euler identity p-q + r = 2-2a yields a = 2, so we have S = 52. Then x(S) = 31#(S2) = "62 = 2 - 2b, so that b = 32 and the covering imbedding of GA(31) is on S = 532. The 31 vertices above are the points of PG(2, 5). The 31 lifts of the hexagon R (p~](R)) depict the lines of PG(2,5). Thus if we take so that \(pm - l)x = 0 in Zv/3. But (\(pm - 1), iv) = 1, so (\(pm - I))" 1 exists in Zw/3, and x = 0 in Zw/3. Thus x = 0 or ±u/3 in Zv. Now we split into our three cases. (1) Case 0. For pm = 0 (mod 3), we have (p7™ +pm + l,pm -l) = 1; thus by Lemma 3.5, either x = 0 or |Fx| = 3. Then the orbit decomposition is:
into one singleton and 3m~] triples. (2) Case 1. Now (p2m +pm + l,pm-l) L
= 3; thus by Lemma 3.6 we have
{ } { }
U
where Si,s2e {0, Ju, 3i»} and all other orbits have length 3. (3) Case 2. For this case (p2™ +pm + \,pm - 1) = 1 again, and Lemma 3.5 yields (p
L=
)
U IX. 1=1
Note that now all orbits have length 3, and 0 $ L in this case. We comment that it is not difficult to show that L is unique, for Cases 0 and 2; it is the unique difference set (line of PG(2, n)) L = {s,-}f having 2?= i $, = (). However, in Case 1, L + \v and L + \v are fixed by p, as L is; precisely one of these three has elements summing to 0 in Zv. Consider the array sx s2 x, pmx, p^x, x2
pmx2
p2mx2
EFFICIENT IMBEDDINGS OF FINITE PROJECTIVE PLANES
45
arising from the orbit decomposition of L under the action of F = (pm) in Case 1. In Case 0, 5! = 0 and we delete s2. I n Case 2 we delete both Si and s2. In all three cases, ^ = V\{pm + 1)1 We will reorder L by the columns of this array in each case, and form our generating set A for Zni+n+l from successive differences in the new ordering. This in turn will lead to a KVL voltage graph, in a natural way. This concludes our algebraic preparation. Now we give the voltage graph constructions, by the same three cases. But this time we begin with Case 2, as the orbit decomposition of s is uniformly into triples for this case. We will see that this is the optimal situation. (1) Case 2. Our array for L is x2
pmx2
"S
r
p2mx2
S
r
S
We reorder L to {a,}? (k = 3s), where Xj
if 1 =5 i < S,
pmXi-s
if s + 1 =£ i =s 2s,
Now set 8j = ai+i— ah with l ^ / ^ 3 s , taking subscripts modulo 3s, and set A = {5J*. 3.7.
LEMMA.
2f=i «,-= 0.
Proof. We have Sf-i 8, = 2f-i ifll+l - a,-) = 0. 3.8.
LEMMA.
5, + 8s+i + 625+/ = 0-
For l^i^s,
Proof. For 1 ^ / *s s - 1, = (x/+1 -Xi) + (pmxi+1 -pmx,) + (p2mxi+, = jci+1(l +/? m +p2m)-xi(l +pm+p2m) = 0.
-p2mx,)
Moreover, 8S + Sis + 83s — (^5 + 1 ~ Os) + (^25 + 1 ~ a2s) "*~ (al ~ ^3s)
= {pmx, - xs) + (p2mXl -pmxs) + (x, -p2mxs) = x](pm +p 2 m + 1) - x 5 ( l +pm +p2m) = 0. We take as our imbedded voltage graph an (n + l)-gon labelled clockwise with
46
ARTHUR T. WHITE
(5,, 82, ••-, 8k), where k =3s = n + 1, with triangle Th where l ^ / ^ 5 , labelled counter-clockwise with (5,, Ss+h 625+,), attached to the (n + l)-gon along the directed edge labelled 5,. See Fig. 3, for an illustration of this. By Lemmas 3.7 and 3.8, the KVL holds. By Theorem 2.3((i) and (ii)(b)) we have an imbedding of G±(Zn2+n+]) above with n2 + n + 1 (n + l)-gons and all other regions triangular. Using (ii)(b) again, we see that if R is the single (n + l)-gon below, then p~\R) = {L-}o7+n. That is, the n2 + n + l lifts of R depict the n2 + n + l difference sets, and hence the n2 + n + 1 lines, of PG(2, n). We obtain an imbedding of G(PG(2, n)) from this Cayley graph imbedding by inserting a vertex in the interior of each L, and joining that new vertex by an edge to each of the n +1 vertices in the boundary of Lh and then deleting all edges of GA(Zn2+w+1). As each edge, bounding a triangle, is thus replaced by a path of length 2, the triangle becomes a hexagon. These hexagons are the only regions of the G(PG(2, n)) imbedding, so that the upper bound of Theorem 3.2 is attained. We will show that the imbedding constructed is orientable, and into a surface.
As each edge below bounds two regions, but in opposite senses, the imbedding below—and hence the imbedding above—is orientable. To show that the imbedding below—and hence the imbedding above—is into a surface, we look at the permutation p of A* = A U A"1 induced by the clockwise ordering of directed edges leaving vertices below. The voltage graph imbedding is into a surface if and only if p has exactly one orbit, in its action on A*. Select the vertex, call it u, at the tail of 8}; the edge carrying 6j separates a triangle and an (n + l)-gon. The clockwise rotation at u begins (~8-2s+\, 8\, -f>3s> •••) and continues at a vertex which we label y on the (n + 1)gon (-5^+1, 8U -83s, 8S, ~8S-U 5^-1, ...)• We note that u and y are separated by a clockwise path of length s - 1 lying entirely on the (n + l)-gon. Continuing this analysis (there are five cases to consider; we omit the details), we find that successive visits to the (n + l)-gon (with the vertices on the triangles not on the edges 5 , ( 1 ^ / ^ 5 ) interspersed) are always a clockwise distance 5 —1 apart on the (n + l)-gon. Now if 5 =0 or 2 (mod 3), that is, if n = 3s - 1 = 5 or 8 (mod 9), then 3 does not divide 5 - 1 , so that (35, 5 - 1) = 1 and there is just one vertex (u = v = ...) below, corresponding to the one orbit of p. If, on the other hand, 5 = 1 (mod 3), that is, if n = 2 (mod 9), then 3 does divide s - 1, (35,5 - 1) = 3, and we have three vertices (orbits) below. (Each has degree is, although this is not crucial.) This produces a pseudosurface imbedding; as we prefer surfaces, we make an adjustment. The three orbits of the action of p on A*, for the case n = 2 (mod 9), are: (~&2s+\,
S i , -82s,
•••),
(-82S+2,
52, - 6 ] , SJ+1,...),
(-8S+U
82S+],
•••)•
Triangle 7, currently has counter-clockwise ordering (8U 8S+U 82s+])- We change this to (Si, 82s+\, 8S+1), by interchanging the edges carrying 8s+l and 8z%+]. This does not affect either the KVL property for 7j (since Zn2+n+] is abelian) or the orientability of the space (the arrows are unchanged). But now the three orbits merge into one:
and we have a surface imbedding. The characteristic below is % = p - q + r = 1 - (n + 1) + (1 + %(n + 1)) = \(2 - n), so by Theorem 2.3(ii)(a) we have % = \(2 - n)(n2 + n + 1), establishing:
EFFICIENT IMBEDDINGS OF FINITE PROJECTIVE PLANES
3.9.
THEOREM.
47
For n=2 (mod 3), y(PG(2, n)) = \{n{n2 - n - 1) + 1).
(2) Case 0. Our array for L in this case is: 0 *1
x2
2m m p xx P X\ m 2m p X2 P x2
m
2m
p x p xs Reorder L by columns, just as in the previous case, except that now the first column has one more entry. We proceed with our voltage graph imbedding just as before, except that we replace triangle Tx with a quadrilateral Qu having counter-clockwise ordering of directed edges (x, - 0, 0 - p2mxs, p2mx} - pmxs, pmxx — xs). These group elements sum to 0, so that the KVL holds for Qx. The calculations used for Case 2 show that the KVL also holds for the (n + l)-gon and for the triangles Th where 2 «= i ss s. Just as in Case 2, the imbeddings above and below are both orientable. But the covering imbedding must be into a pseudosurface (not a surface), as X
X = (n2 + n + 1) - (n + l)(n2 + n + 1) + (\n + l)(n 2 + n + 1)
= ( l - | n ) ( « 2 + n + l) is odd. There is no remedy for this! In conjunction with Theorem 3.2, we have shown, for this case, that (1 - ln){n2 + n + 1) *s^'(PG(2, n)) ^ (43 - \n){n2 + n + 1), and the bounds agree, asymptotically. 3.10.
THEOREM.
Forn=0 (mod 3), #'(PG(2, n))« 1(2 - n)(n2 + n + 1).
In Fig. 4 we show the case n = 3. The 13 lifts of the unshaded region are the 13 lines (hyperedges) of PG(2,3); the 13 lifts of the shaded region are the hyper-regions. The orbits of p are (2,-1,4, -6) at W and (6, - 2 , 1 , - 4 ) at Y; this is unavoidable. The imbedding of PG(2,3) is into 5(1; 13(2)), a torus with 13 singular points: (W, i) is identified with (Y, /) above, for each i in Z ]3 .
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ARTHUR T. WHITE
(3) Case 1. Now our array for L is:
x2
pmx2
p2mx2
Our analysis proceeds as in Case 0, except that now we replace 7, (or Qx) with a pentagon, />,: (*, - su 5, - p2mxs, p^x,
- pmxs, pmxx - s2, s2 - xs).
Again the KVL holds globally and again orientability is immediate. Our verification that we can achieve a surface imbedding for this case is similar to that for Case 2, except that here we focus our attention on the vertices M, of the triangles Tt not immediately incident with the edge carrying 8h for 2 =s / ^ s. In the orbits of p on A*, we find that the number of directed edges between successive w, in a common orbit is: 6, from u2 to us\ 8, from u3 to w5_2 and from u4 to us-i\ and 4, from uh to uh^, for 5 ^ h *£ s. It follows that there is just one orbit of p, if s = 0 or 1 (mod 3), that is, for n = 3s + 1 = 2 or 4 (mod 9), and three orbits (of lengths 2s, 2s, and 2s + 4, althouth this is not crucial) if s = 2 (mod 3), that is, for n = 7 (mod 9). In the latter case, re-ordering P by interchanging S\-p2mxs and pmxl -s2 combines the three orbits into one, in a manner similar to that of Case 2, without affecting either the KVL or orientability. Thus we have an orientable surface depiction of PG(2, n), for n = 1 (mod 3), in each of the three subcases. We check that % = (1 - (n + 1) + (1 + i(n - l)))(/i2 + n + 1) = ( - § « + \){n2 + n + 1); thus (5 - h)(n2 + n + l) «s *'(PG(2, /i)) «£ (J - §n)(/72 + n + 1), which yields: 3.11.
THEOREM.
Forn = l (mod 3), y(PG(2, n)) -= K«(«2 - n - 1) + 1).
The asymptotic result *'(PG(2, «)) « 1(2 - n)(n2 + n + 1) is the strongest claim that we can make for x' for all prime powers n, but we reiterate that for n = 1 or 2 (mod 3) the imbeddings are into surfaces, and for n = 2 (mod 3) the result is exact. Moreover, even for n = 0 or 1 (mod 3), the results are exact in the sense that, under our proviso that Zn2+n+] act regularly on each region orbit (so that if we have one quadrilateral above, we must have at least n2 + n + 1 quadrilaterals; if we have one pentagon above, we must have at least n2 + n + 1 pentagons), the imbeddings constructed attain the corresponding upper bound for x'r- ^n summary, then, for all three cases we have: 3.12.
THEOREM.
For i = 0,1, 2 and n=2-\- i (mod3),
X'r{VG{2,n)) = \{A-i-2n)(n2
+ n + \).
We close this section by commenting briefly on the geometric nature of the
EFFICIENT IMBEDDINGS OF FINITE PROJECTIVE PLANES
49
hyper-regions we have constructed, and on the map automorphism groups of the imbedded Cayley graphs. For /2 s 3, an l-arc in PG(2, n) is a set of / points, no three of which are collinear. Since each vertex is incident with n + 1 distinct lines (and n + 1 hyper-regions), three consecutive vertices in a hyper-region will never be collinear. Thus, in our covering imbeddings, every 3-gonal hyper-region is a 3-arc (called a triangle for PG(2, n)), and every 4-gonal hyper-region is a 4-arc (called a quadrilateral for PG(2, n)). Low-order examples suggest that all 5-gonal hyper-regions (occurring only for n = 1 (mod 3)) are 5-arcs, but (as three vertices of a pentagon can be non-consecutive) the simple argument given above no longer applies. Let M{nl) denote the imbedding we have constructed for PG(2, n) in this section; it is of the Cayley graph GA(Zn2+w+1) for our carefully chosen generating set A into its ambient (pseudo)surface. The superscript reminds us that A = 1; in § 5 we will construct M(2) for A = 2. By Aut M^ (i = 1, 2) we mean the group of all permutations of the vertex set T- Zn2+n+} of GA(F) which preserve oriented region boundaries. (Since such permutations necessarily preserve adjacency, Aut M^ss Aut GA(F).) It is convenient to define the permutations cr, and nk, where /, k e Zn2+n+] and (n2 + n + 1, k) = 1, by (1) aj(x) = x+j, and (2) nk(x) = kx.
In general, it can be quite difficult to calculate Aut A/*,0; we shall do so precisely only for / = 1 in Cases 0 and 1 and for small values of n in the other cases. Let txi denote the semi-direct product operation for groups. For the remainder of this section, i = 1. Then AutM 3), we have a 2/-gon region for G(D), and this gives all the regions for G(D). (To visualize this, it might be helpful to examine Fig. 2(b). Also, consult Lemma 5.6 of [21].) We again consider the three cases, modulo 3.
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ARTHUR T. WHITE
(1) Case 2. The Cayley graph imbedding has rn + ] =n2 + n +1 and r3 = (n + n +l)(/i + l)/3; thus the G(D) imbedding has r = r6 = (n2 + n + 1) X (« + l)/3. Since G(D) has girth 6, this surface imbedding attains the genus of G(D). (2) Case 0. The Cayley graph imbedding has rn+l = n2 + n + 1, r3 = 2 (/i + n +l)(n - 3 ) / 3 , and r4 = n2 + /i + l. Thus the G(D) imbedding has r6 = (n2 + M + l)(n - 3)/3 and r8 = n2 + n + 1. This pseudosurface imbedding is easily seen to be asymptotically efficient, for the pseudocharacteristic #'(G(PG(2, n))). (3) Case 1. The Cayley graph imbedding has rn+l = n2 + n + 1 , r3 = 2 (n + n + l)(n - 4)/3, and r5 = n2 + Ai + l. Thus the G(D) imbedding has rb = (A?2 + « + l)(n - 4)/3 and r]0 = «2 + n + 1. This surface imbedding is easily seen to be asymptotically efficient, for the genus of G(D). Let Gn denote the (n + 1, 6)-cage associated with PG(2, n). Then we have shown: 2
4.5. THEOREM, (i) For n =2 (mod 3), y(Gn) = \(n{n2 - n - 1) + 1). (ii) For n = 1 (mod 3), y(GJ * \{n{n2 - n - 1) + 1). (iii) Forn=0 (mod 3), x'(Gn) - |(2 - n)(n2 + n + 1). Letting /? = 3w - 1, we write (i) of Theorem 4.5 as: 4.6. COROLLARY. For each w such that 3w - 1 is a prime power, y(G3M,_1) = w(3w-2)2. The smallest value for Corollary 4.6 is w = 1, and that brings us back to the 'Heawood Map' of Fig. 2(b). 5. The case A = 2
Here we depict PG(2, n) and the inverse plane -PG(2, n) simultaneously. The corresponding design is a (v, b, r, k, A)-BIBD with parameters (n2 + n + 1, 2(n2 + n + 1), 2(n + 1), n + 1, 2). We denote this design by D = 2PG(2, «), since the two 1-fold designs are isomorphic. We observe that -PG(2, n) is the dual of PG(2, n) (see [13]). Our constructions will attain the euler upper bound for pseudocharacteristic (not just asymptotically), will always be orientable, and will be surface imbeddings if and only if n is even (p = 2). The bipartite incidence graph G(D) for D = 2PG(2, n) has (n2 + n + 1) point vertices of degree 2(n + 1), and 2(n2 + n + 1) line vertices of degree n + 1, and hence has order p = 3(n2 + n + 1) and size q = 2(n + l)(n2 + n + 1); the girth is 4. (Since A = 2, 4-cycles of the form (P\> 1\>P2> h) abound, where /, and l2 are the unique two lines that px and p2 + r : belong to.) Thus 2g2=4r, and r *£ (n + l)(n2 + n + 1). Hence X=P~(i ^ 2 2 (3 - 2(n + 1) + (H + \)){n + n + 1) = (2 - n)(n + n + 1). In summary: 5.1.
THEOREM.
*'(2PG(2, n)) ^ (2 - n)(n2 + n + 1).
We will use the same Cayley graphs G&(Zni+n+l) as in §3, although any A formed by taking successive differences from an arbitrary ordering of any difference set for PG(2, n) would suffice.
EFFICIENT IMBEDDINGS OF FINITE PROJECTIVE PLANES
53
Again, for our voltage graph imbedding, we start with an (n + l)-gon, labelled in the clockwise direction with generators (8U 82,..., 8n+\). But now we attach only a second (n + l)-gon, with clockwise labelling (-8U -82,..., -8n+i), along the edge labelled 5) (say). See Figs 2(c) and 4 for the cases n = 2 and 3 respectively. (These two voltage graph imbeddings, and the one for n = 4, all do double duty—since (n + l)-gons are also triangles, quadrilaterals, and pentagons respectively, for these cases.) Identify the remaining edges, as indicated, to obtain the voltage graph imbedding. The KVL is satisfied on both (n + l)-gons, by Lemma 3.7, so the first will lift to the lines of PG(2, n), and the second to the lines of -PG(2, n). As each edge appears once in each sense (in a consistent clockwise direction), the imbedding below (and hence above) is orientable. It is straightforward to check that p = 1 for n even, but p = 2 for n odd. The usual modification to imbed G(D) above (now a vertex is inserted in every covering region) produces f = r4, so all imbeddings are optimal. See also Theorem 5.3 of [21]. We have shown: 5.2. THEOREM. (1) For n even, y(2PG(2, n)) = \n(n2-n(2) For all n, *'(2PG(2, n)) = (2 - n){n2 + n + 1).
1).
The same formulae apply to G(D), but, as we have seen, these graphs are not regular; hence they are neither distance-regular nor cages. Thus they seem to have less interest, in their own right, than their counterparts for A = 1. Let cru ;!•(_!), Kp, and Kpm be as in §3. Let M(2) be the map for 2PG(2, n) constructed in this section. Then our voltage graph constructions, by Theorem 2.3(ii)(c), guarantee that crx acts on M{2\ so that Zn2+n+i *£ Aut M{2). It is also immediate that K{-\) acts on M{2), with n2 + n +1 region orbits, each of size 2. In fact, 7T(_j) e (AutM^2))0, so in each case Zn2+n+1t>cZ2^ Aut M{2). Using the fact that Kp" (with w = 1 or m) is in Aut M(2) if and only if KP- cyclically shifts ordered generators on every region (Lemma 3.14), we can show the following. (1) For Cases 0 and 1, n^ e (Aut M(2))0> so that also np $ (Aut M(2))0. (2) For Case 2, Kpm e (Aut M{2))0, so that Zni+n+i ixZ 6 ^ Aut M(2). (3) For Case 2 and m > 1, KP E (Aut M(2))0 if and only if n = 8. Thus Aut M82) = Z 73 txZ, 8 , where Z18 =
