A duality theorem for graph embeddings

A Duality Theorem for Graph Embeddings

Brad Jackson T. D. Parsons’

Tomai Pisanski PENNSYLVANIA STATE UNIVERSITY UNIVERSITY PARK, PENNS YL VA NIA

ABSTRACT

A generalized type of graph covering, called a “wrapped quasicovering” (wqc) is defined. If K,L are graphs dually embedded in an orientable surface S, then we may lift these embeddings to embeddings of dual graphs z,z in orientable surfaces g, such that g are branched covers of S and the restrictions of the branched coverings to z,r are wqc’s of K.L. The theory is applied to obtain genus embeddings of composition graphs GfnK,] from embeddings of ”quotient“ graphs G.

1. INTRODUCTION

In a previous paper [lo], we introduced the notion of “wrapped quasicoverings” of graphs, and stated without proof some theorems which we then applied to obtain genus embeddings of certain families of graphs. We shall now prove those theorems, and give some further applications. We assume that the reader is familiar with the theory of graph embeddings, particularly the use of current graphs and voltage graphs. (Our references include selected papers which supply the necessary background [ 2,3,4,6,11,15]).

Suppose that K, L are dual current and voltage graphs (respectively) in an orientable surface S, where the currents and voltages are in some finite group I?. Let z be the usual derived (regular-) covering graph of the voltage graph L, in the sense of J.L. Gross L3]. It was Gross who first discovered the relationship (Fig. 1 .), where X: L -, L is the natural covering map of E over L andB: g -, S is a branched covering of surfaces such that the restriction of B to z is A. In fact, Gross and Alpert [4] first discovered the general theory of current graphs as it relates to branched coverings of surfaces, where one goes directly from the current graphK to the “derived” embedding o f E in g

Research partially supported by N.S.F. Grant MCS-8002263.

Journal of Graph Theory, Vol. 5 (1981) 55-77 0 1981 by John Wiley & Sons, Inc. CCC 0364-9024/81/010055-23$02.30

56 JOURNAL O F GRAPH THEORY

F IGURE 1

(Fig. 2). Here, regions of in 9 cover vertices o f K , perhaps with wrapping, and vertices of degree d in E cover regions with d edges of K in S.

Shortly thereafter, Gross invented voltage graphs to supply the covering of L “over the dual” L ofK in S, thereby obtaining the more useful information of our Figure 1. Voltage graphs became important immediately, since they often gave a better (simpler) way of embedding E in 9 by lifting the embedding of a suitable quotient L in S. In this case, vertices of degree d i n E cover vertices of degree d in L , and regions o f E in 9 cover regions o f L in S , possibly with wrapping. Although Figure 1 was initially used for “regular” coverings arising from voltages in groups, Gross and Tucker [ 6 ] then extended it, via generalized “permutation voltages” to arbitrary coverings X :

in 9 (Fig. 3) then usually I? will not be a cover of K (it will fail to cover K precisely when the Kirchoff Voltage Law fails for L in S ) . Instead, it will be a “generalized cover” of K. This is the point of departure for our theory ,Of “wrapped quasicoverings” (wqc’s) w : - K. In this theory, vertices o f K cover vertices ofK, possibly with wrapping, while regions with d edges of in 9 cover regions with d edges of K in S . We shall seek an appropriate notion of “generalized

L-+ L. If, in Figure 1 , we consider the dual g of

FIGURE 2.

DUALITY THEOREM FOR GRAPH EMBEDDINGS 57

I b ‘ W J.

FIGURE 3.

covering” so that Figure 3 will hold with both o : I? -. K and X : ,f - L as generalized coverings which are the restrictions of the branched covering B to Z? andL , and where w “completes” Figure 1 for coverings X :

As an example, look at the graphs in Figure 4, where the current graph K and the voltage graphL have their respective currents and voltages in Z2, and where all the graphs are embedded in the sphere. The mapping X from ,f to the voltage graph L takes vertices to vertices, edges to edges, and regions to regions with wrapping. The wqc o also takes vertices to vertices, edges to edges, and regions to regions, but the wrapping now occurs about the vertices.

4 L.

L

a

FIGURE 4.

58 JOURNAL OF GRAPH THEORY

Now consider the following special case of Figure 3. Suppose that w : a - K is a wqc and that b o t h a and K are triangularly embedded in the surfaces 3 and S , respectively. Then a and K have embedding schemes. (See Ringel’s book [ 1 11 for the definition of an embedding scheme.) Also suppose that, for each vertex v E K and vj E w-’(v) , in the embedding scheme for K we have that row v is uoul * *urn and in the embedding scheme for we have that row vj is uyu: * - u;” uiui * * * ~ 8 . ~ 1 . - * * ut., where o(u i ) = ui, for all i,p. That is, row vj is mapped cyclically onto row v exactly Sj times via o. Then the embedding scheme for i?: is said to be a “wrapped embedding scheme” lifted from the embedding scheme of K, and Sj is the “wrapping index” of vertex vj over vertex v .

One particularly nice example of a wqc of a graph G is the composition of G over m independent vertices; this composition is denoted by G[mK,] or G(,). The graph Gtm, is obtained from G by replacing each vertex u of G by rn mutually nonadjacent vertices u1,u2, - - * ,u, and by replacing each edge (u,v) of G by the set of ( ui,vj) for all i j = 1,2, * * ,m. The mapping w : G,,, -+

G defined by w(ui) = u and w((ui,qi) ) = (%ti), is a wqc with wrapping index m at each vertex of G(,,.

In [ I ] Bouchet gave a method using “m-valuations” for generating triangular embeddings of G(,) from triangular embeddings of an Eulerian graph G. His method worked for all integers rn to give wrapped embedding schemes for G(,, from an embedding scheme for a complete multipartite graph G. He then used this method to construct many triangular embeddings of the complete multipartite graphs Kicm, with i parts of size m. Other methods of constructing wrapped embedding schemes are discussed in papers [8,9] by one of us (Jackson).

In the present paper, we do not need to assume that G is Eulerian in order to obtain wrapped embedding schemes for G(,,. We are able to show that, for any graph G which is triangularly embedded into an orientable surface, there exist infinitely many values of m for which Gtm, can be triangularly embedded into an orientable surface.

Finally, we would like to thank the referees for their many helpful suggestions.

’

J J

2. WRAPPED QUASICOVERINGS

Agraph K is an oriented 1-dimensional cell complex in which each open edge (open 1 -cell) has a preferred direction. (Therefore, loops and multiple edges may occur.) An oriented edge is a directed 1-cell having either the preferred direction or the opposite of the preferred direction, and e-’ denotes the oriented edge directed opposite to e. If K has no loops, and at most one


edge contains both of any pair of distinct vertices, then K is a simplicia1

Agraph homomorphism h: J - K is a continuous surjective mapping which maps the vertices of J onto the vertices of K, and maps each open edge ofJ homeomorphically onto some open edge ofK. It is direction preserving if it preserves preferred directions of edges.

Let g,K be graphs, with K connected and g having no isolated vertices. A wrapped quasicovering (wqc) w: -, K of over K is a direction preserving graph homomorphism such that

(1) There is a positive integer N (called the mult@licity of w ) such that for every open edge e of K there are exactly N open edges of a in the preimage w-’(e) of e.

(called the wrapping index of w at F ) such that, if v = w(F) then for each edge e of K with preferred direction incident to (respectively, incident from) v , there are exactly 6 ( F ) edges o f g in w-’(e) incident to (from) i7. (A loop at a vertex is counted as having its preferred direction both to and from the vertex.)

Note that if w - ’ ( v ) = {TI , . * * , T r } , (where r depends on v ) , then 6 ( F l ) + * * 4- a(;,) = N, the multiplicity of w. If some S(Fi ) > 1 for F i f w-’(u), then v is a singular vertex of K.

If 6 ( F ) = 1 for every vertex F o f f , then w is a covering map in the sense of Gross and Tucker [ 6 ] , and the restriction of w to each component ofK is a covering in the usual sense of topologists.

Example 1. Figure 5. Here N = 6 , 6 (Ll ) = 4, 6(L2) = 2, a(;,) =

graph.

(2) For each vertex F of g there is a positive integer

2, 6(F2) = 1, 6 ( F 3 ) = 3.

If w: -, K is a wqc and F is a vertex of with g(i7) > 1, then we may “split” i7 into new vertices 8 ’ 1 , * * * ,F(@ (6 = 6(v )) and identify these new vertices with the endpoints of the edges F incident to or from i7, so as to obtain a new wqc in which S(F(’3) = 1 for i = 1 , - - , 6(5). When this is carried out for every such vertex 5’ ofg, we obtain a new wqcj: J -, K which is a covering map in the sense of Gross and Tucker. (Usually there are many different new wqc’s obtainable in this way.) For example, we may obtain from the wqc of Example 1 the following covering map:

Example 2. Figure 6.

A different J could be obtained by replacing the last two components of the one above by a single component of the form in Figure 7.

A special type of homomorphism is a vertex identification, which “identifies” certain vertices of a graph into single new vertices (without


FIGURE 5.

changing the number of open edges). By reversing the process which “split” a into J, we obtain a vertex identification i: J - a such tha t j = wi . Conversely, given a covering mapj: J - K and a vertex identification i:

J - if such that identifications occur only within the fibersj-’(v) over vertices v ofK, it is clear that w determined by j = wi is a wqc w: i? - K.

Therefore wqc’s may be viewed as special homomorphic images of coverings.

Theorem 1. A hormomorphism w: K - K is a wqc if there is some covering j : J - K and some identification i: J - of subsets of the fibersj-’(v) o f J over vertices v of K , such tha t j = wi.

N

I

We now turn to our main result, which involves the notion of “lifting” an embedding of a graph K in an orientable surface S to an embedding of x in another orientable surface s, by means of a derived “wrapped” rotation on a relative to wqc w: x - K. We first note that since w is direction preserving, it induces a mapping (also denoted by w) on the set of oriented edges o f k onto the set of oriented edges ofK, such that o(C’) = [w(F) ] - - ’ .


K FIGURE 6.

FIGURE 7.


A rotation p on K is a permutation on the set of all oriented edges of K such that each cycle p, of the disjoint cycle decomposition of p corresponds to a vertex Y of K and is a cyclic permutation of the set of all oriented edges out ofu. The permutation p induces another permutation p* on the set of oriented edges of K, defined by p*(e) = p(e-’). The cycles in the disjoint cycle decomposition of p* correspond to the oriented boundaries of the 2- cells in a cellular embedding of K in an orientable surface S. Although the manner in which this embedding is determined by the rotation is well known (see [ 111) we shall discuss the construction in some detail in order to give a convincing proof of our main theorem.

If (el ,e2, - . * ,ek) is a cycle in the disjoint cycle decomposition of p*, then we construct a corresponding unit disk D in the complex plane with k equal arcs on its boundary set off by the k vertices exp (i2nj/k) for j = 0, 1 , - * ,k- 1. Starting with the vertex z = 1 = exp (i277*Olk), we proceed counterclockwise around the boundary of the disk, consecutively labeling the arcs as el ,e2, * ,ek. (Here the “starting” arc el is essentially arbitrary, since (e , , - . * ,ek) is a cycle.) We give D itself the counterclockwise orientation, and we let i(e,) and t(e,) denote the initial and terminal vertices of the (counterclockwise directed) arc e, on the boundary of D. The oriented boundary ofD is thus dD = (e1,e2; * *,ek) .

If D(e) is the disk containing oriented edge e in its boundary, then some disk D(e-’ ) [possibly D(e) = D(e-’)] contains e-I in its oriented boundary, Let 0, be the homeomorphism from the circular arc (i(e), t(e)) on dD(e) to the circular arc (t(e-’),i(e-’)) on dD(e-’) with the following interpolating property: 6,(i(e)) = t(e-’), and if x is the point on arc (i(e), t(e)) which is located with fraction s(0 S s S 1) of the total arc length from i(e) to t(e), and y is the point on arc (t(e-’) , i(e-’)) such that arc (t(e-’),y) has the same fractions of the total arc length from t(e-’) to i(e-’), then &(x) = y . Note that 0,-’ = (0,)-’. We now identify each point x on arc (i(e), t(e)) on dD(e) with the point &(x) on aD(e-’) in Figure 8.

The topological space arising from making these identifications on the boundaries of all the disks D(e) by the “attaching maps” 6, is an orientable surface S in which K is embedded so that the complement of K in S is a family of open 2-cells, namely the interiors of the disks. - K is a wqc. We may suppose that and K are represented by copies in which the edges are rectifiable curves in some Euclidean space, such that each edge is the homeomorphic image of a nondegenerate closed interval on the real line, and that if i? is any edge i n x from iT = i(F) to F = t (F ) (possibly t = F ) , and 2‘ is the point at fractions (0 5 s I 1 ) of the way along F from i7 to F , then ~(2’) = x is the point at fraction s of the way from u = o(E) to v = o(F) along e = w ( F ) in K. That is, we may assume without loss of generality that o:

Now suppose that x is connected, and that o:

- K has the “interpolating property.”


FIGURE 8.

A rotation F on x is wrapped relative to w and the rotation p on K , if w F = pw. The geometrical effect of F at a vertex i7 o f x is to project via w so as to "wrap around" the rotation of p at

Here w(iTi) = a, etc., and b = w(g i ) = w(F(&,)) = p(o(Zi ) ) = p(a), etc. If i7 is wrapped, relative to w and p, then we also have w F * = p*w. Indeed,

if a(;) = e, then wF*(Z) = w F ( E - ' ) = pw(Z- ' ) = p(w(F)- ' ) = p(e-') = p*(e) = p*o(F).

Let p be a rotation corresponding to an embedding of K in S , let w;i? - K be a wqc, and let F be a rotation on i? which is wrapped relative

= w(F) in K (Fig. 9).

1-

FIGURE 9.


FIGURE 10.

to w and p. We shall illustrate how w p = pw and wg* = p*o correspond to a wrapping of the surface ,!? (in which p embeds K ) over the surface S .

under the action of p* is the oriented boundary of a face ofa i ng . Since up* = p*w, the projection via w of such an oriented boundary in I? “wraps around” a corresponding oriented boundary in K (Fig. 10).

Now suppose (e1e2, * - ,ek) is the oriented boundary of a face D(el)of the surface S constructed from the rotation p on K , and that the initial point i(el) corresponds to the complex number z = 1 on D(el). Let Fl be any oriented

Each cyclic orbit of an oriented edge i? of


edge o f a such that o(Zl) = el. We may assume that the initial point i (FJ of Z, on the disk D(Zl) also corresponds toz = 1. Since the boundary of D(Zl) wraps around that of D(el), via projection under o, we have that, for some integer d h 1 , the mapping z -, zd of the complex unit disk maps D(Zl) onto D(e,), and this mapping reduces to w on the b o u n d q o f D(Z,) by our previous assumptions on the interpolating property of o: K -. K, along with the fact that z - zd has the same interpolating property on arcs of radian measure less than 27dd on the boundary of the unit disk. [Of course, ifD(Zl) wraps around D(el) d times, then each arc Z in dD(Zl) has radian measure less than 27dd, Gnce dD(Zl) must contain at least d equal arcs.]

Defining B: S -+ S on the interior of each such disk D(Zl) by the appropriate mapeng z 2 zd onto the corresponding D(el ), znd defining B(2) = ~(2') for 2' E K on S ,we have that for all pointsy E S such that is

FIGURE 11.


not a vertex of K" nor a center of a disk D(Z), B maps a neighborhoodzf 7 homeomorphically onto a neighborh_ood ofy = B F ) on S. Therefore B: S -, S is a branched covering of surface S over S, (sLe RH. Fox [ 2]), with branch points contained in the union of the vertices of K with the centers of the disks q Z ) . The singular points of S relative to B occur, therefore, among the vertices of K and thesenters of the faces D(e) of K in S.

If i7 is a vertex of K with wrapping index &C) over v = , o ( C ) in K, +en w 6 = pw and WE* = p*w imply that a neighborhood M of F on S is m_apped to a neighborhood M of v on S so that the restriction of B to M\(C) is a &?')-fold covering thus if a(?) > 1, C is a branch point of branching index &C), but if 6 ( C ) = 1 no branching occurs at i7. This is illustrated in Figure 1 1 with S(Q = 2 amd v of yalency 3.

We now discuss the duals L,L gf K,K on S,S . For each disk D(e) and q Z ) in the construction of S and S , construct a vertex in the center of the disk-and draw a line segment from it to the midpoint of each oriented edge of K, K in the oriented boundaries of D(e), D(Z). SucJh a linLsegment, from center to midpoint, represents half of an edge of L,L on S,S . If o(Z) = e, then B maps D(E) onto w e ) via a mapping z + zd which takes the ce2ter vertex C of D(Z) to the center vertex w of D(e) and the "half-edges" of L in D(F) to those in D(e) as illustrated in Figure 12, for a case with d = 2 and the number of edges in aD(Z), aD(e") equal to 6,3.

The midpoint of F is sent to the midpoint of e by B, and the ray from i? to g e midpoint of e" is sent to the ray from w toLhe midpoint @e. It follows that L = B-'(L) and the restriction of X ofB to L is a wqc A: L + L, where the wrapping index of C E A-'(w) over a Zrtex w of L is the same as the branching index 0,f in S relative to B:S -, S. Modifying the embedded grThs K,L and K,L by suitable homeomorphisms applied to the surfaces S,S constructed above, we conclude our main result. (See Fig. 13.)

Theorem 2. Let K" be connected and let o: -, K be a wqc. Let ,K be embedded in an orientable surface S-by the rotation of p, and let K be embedded in some orientable surface by anx rotatie F which is wrapped relative to w and p . Let the duals L,L zf K,K in S,S be embedded in_the usual manner, so that each vertex of L,L corresponding to a face of K> is represented by an interior point @ that face, and edges of L,L are repLesented by simple curves on S,S crossing the corresponding edges of K,K_ at a single point interior to the edge. Then there is a breched covering B: S -, S of surfases such t h a t l l ) The restriction of B to K is w. (2) The restriction of B to&Ls a wqc k L - L. (3) The branch points of B occur at those vertices of K ,L which map to the singular vertices of K,L (relative to the wqc's w and A), and their branching indices are equal to their wrapping indices. 1


e3

FIGURE 12.

REMARK 1. Although the duals LYE of K,E in S,s" were constructed geometrically above, they can also be defined combinatorially, as is well known in the folklore of graph embeddings. One takes the vertices of L, for instance, to be the cyclic orbits of p*, and one labels the edges of L with the same names as the (corresponding) edges of K with preferred directions as follows: if e is an oriented edge of K with the preferred direction in K, and if e occurs in orbit w1 of p* and e-l occurs in orbit w, of p*, then L has a (preferred direction) edge named e from initial vertex w, to terminal vertex w2. (See Fig. 14.) It follows that p* can be regarded as a rotation on L, since the disjoint cycles in its decomposition are in correspondence with the


‘L N

5

FIGURE 13.

vertices w ofL and each such cycle p*, is a cyclic permutation of the oriented edges out ofw. Of course, p**(e) = p*(e-’) = p((e-’)-’) = p(e). Thus if L is embedded in a surface S by rotation p*, then the oriented boundaries of the faces correspond to the orbits ofp, i.e., to the vertices ofK, and the dual ofL is K.

REMARK 2. By the duality in_Theorem 2, we could start with a wqc X:Z --t L and obtain from it a wqc w : K -, K. In the special case where X is a covering, we obtain the situation discussed in the Introduction. We shall now return to this situation, for voltage graphs L with voltages in finite groups r. If e is an edge of L having initial vertex u an^ terminal vertex v , and if y E r, then we shall denote by (e,y) the edge of L having initial vertex (u ,y ) and terminal vertex (v,y@), where @ is the voltage assigned to the edge e. Then (e,y) and (u,y) are the “lifts” of e and u at the group element y E r. (See Figure 15).

FIGURE 14.


FIGURE 15.

3. APPLICATIONS TO "EXCESS CURRENT GRAPHS'

Let K,L be dual voltage, current graphs in S, with voltages and currents in the finite group r, and where P denotes the voltage or current of an oriented edge e of K or L. Consider the special case of Theorem_ 2 corresponding to Figure 3 of the-Introduction. We assume_that the cover L of L is connected. It is the graph K andjts embedding in S that interests us. In particular, we shall usually wantK to be simplicial, so we shall want to know what features of the current graph K will imply that k is simplicial.

Proposition 1. If K has no loops, then K" has no loops.

ProoJ: If K" had a loop Z at i7, then o(Z) would be a loop at w(;) in K. i This simple ccndition can be improved, giving necessary and sufficient conditions for K to have no loops.

Proposition 2. Suppose that Z, is a loop at F i n g , and let el = w(Z' , ) and u = w(i7). Let the rotation pv at u in K be given by (e,,e2, - - .ep). Then for some j ( 2 I j 5 p ) and some integer P 2 0 we have ei = el-' and (Clg2* * *Sb)'2122- * = 1 ,the identity of r. Conversely, if the rotation pu at %satisfies these two conditions, then el lifts to loop Z l at vertex F over Y in K .

ProoJ: Recall that we denote corresponding oriented edges, in a pair of dual graphs in a surface, by th_e same names (with ambiguity resolved by context). If F l is a loop at i7 in K , then there is a face D(Fl) of L" in s" such thatD(F,) contains both of the oriented edges Zl,T1 of L" in its oriented boundary. Let el = A(?,) in L. Then the face D(e,) of L in S contains both el, el-' in its oriented boundary. Therefore the corresponding edge eL-of K is a loop at Y, and if py = (e1,e2; - .,e,) in K, we must have ej = e;'for some j ( 2 5 j 5 p ) (Fig. 16). The face D(e, ) of L in S, corresponding to the vertex v of K, has oriented boundary dD(e, ) = (el ,e2, * * ,ep) in L. By the theory of


FIGURE 16.

voltage graphs, the face D ( Z l ) in must, for some y E r, have oriented boundary

dD(.Fl) == ((e,,y),(e,,y~,),- * *,(ep,yel%. - * c p - d , ( e ~ , ~ 4 % * * * g p ) , - * -,(e,,yP)),

where p = (C,e2- - * ~ ~ , ) ' 4 2 ~ e ~ - - -.C?-J and (P1e2* * -I?,)~+' = 1 in r. Since El = (el,y) and?;' = (e:',a) = (ej,a) both are in dD(Z1), we must have a = y(elz2. * ~ ~ ) ~ e ~ e ~ - .ej-' for some integer I Z 0. But E;' = (e1,y)-' = (e:',ye,) = (e.,yel), so also a = ye1. Therefore yel = y(i?,e2. *ep)ri?+2 * *

gjPl. Since Z l i = ei' = Zj, we conclude that (ClP2- * *i?p)rC?lP2 * * = 1.


The converse is easily obtained by reversing the steps of the argument. I Suppose that el is an oriented edge of K with initial vertex v and terminal

vertex w. Let the cyclic rotations at v,w be p, = (el,ez- - -ep) and pw = (el-'&, * - a&). We then define the initial excess current of el to be the group element C1C2 * * mep E r, and the terminal excess current of e, to be 3 * - $c1-'. (Note that if v = w, so el is a loop at u, then the terminal excess current /3 at el and the initial excess current (Y at e;' are related by p = e 1 cue;'. )

Proposition 3. If the initial excess current of el has order 8, in r, and if v is of valency p in K, then w-' (v) contains Exactly lr'l/8, vertices each of valency p * 6, and of wrapping index 8, in K . If D is a face of K in S _with Sactly d orient@ edges in its boundary, then t k r e are exactly I rl faces D of S such thgB(D) = D, and each such face D has also exa$y d oriented edges in dD. Each edge e of K has I rl edges in w-'(e) in K .

Proo$ The initial excess currents of any two oriented edges out of u in K are conjugate in r, and thus have the same order 6, in r. The corresponding "excess voltage" elPz - - Cp arcundifle,) in L has order 6, in r, so $ere are exactly I rl /a, faces D(7') of L in S which project to D(e,) by B: S .-) S, ankeach such face hasp 6, edges in itshouniary. The vertices E E w-' (v) in K correspond to these faces D(7,) of L in S , so there are I rl /a, of them, each of valencyp6, and wrapping index 6,.

A face D of K in S, with exactlyp oriented edges of K in dD, corresponds to a vertex-of valency d in L. This vertex lifts to I rl vertices 0,f valency d in the cover& of $e vgtage graph &. Each of these vertices of L corrEspon@ to a face D of K in S , such that D has exactly d oriented edges of K in dD and projects to D under B: $ -+ S. Each edgEe of K corresponds to an edge e E L , which lifts to 1 r / edges .F of L , each corresponding to an 7 E w-'(e) i n Z . I

Two subgroups of a group are disjoint if they intersect in the trivial subgroup.

Proposition 4. If K is simplicia1 and the initial excess current and the termiKal excess current of each edge e of K generate disjoint subgroups of r, then K is simplicial.

Proo$ has no kops, by Proposition 1. Suppose 7 and 7' are distinct oriented edges of K_, eachJaving initial vertez F a_nd terminal vertex G, where F # C . Let D ; a_nd D; be the faces of L in S which correspond to the vertices E,C of K . Let v = w(F) and w = w(F) in K, and let


N

e

-1 (P)-'

I V l

- L

FIGURE 17.

D,,D, be the corresponding faces of L in S. Consider e = o(Z) and e' = w(Z'). Note that o # w, else e would be a loop in K. If e # e', then K would have e, e' as multiple edges- from o to w. Therefofte e = e'.

We have that Z,Z' E dD; andZ-',(Z')-' E dD;, while e E dD, and eC1 E dD, (See Fig. 17).

Let p , = (el,e2; * -e,) andp, = (el-' ,&,- - ef,) be the cyclic rotations at o,w inK, whereel = e. ThendD,= (e1,e2; * *ep) anddD,= (er',f2; * -,&) in L. By the theory of VoltageJraphs, for some y E r we have in z that aD; = ((el,y),(e2,yel), * * a ) anddD; = ( ( e l ~ l , y e l ) , ~ , y ~ l e l ~ ' ) , ~ - .). Since el = e=e ' , andZ=(e , ,y ) ,wemusthaveF ' = (e,,y(Z,Z,. * Z p ) ' ) for some integer Y > 0 such that (ZIz2 * - *eJ # 1 in r (here we use Z' f Z).

(z'>-'= (e1,y(e,e2 * - -eP>r)-' =(el-1,y(~le2 - . zPf'p,) E dD;; therefore for some s > 0 such that (eTlf2. - -A)' f 1 in G, we must have (e;',yZl(i?;1y2- - -$)') = (e;',y(e1C2. * .Zp)'6). This implies that yi?;'f;* * *

&)s = ( e 1 Z 2 * * *2Jr # 1.But then the initial excess current - -ep) and

Now Z-' = (e1,y)-' = (el-',yZ,) E dE,;, and also N


the terminal excess current cf2 - *&a;’) of el = e in K generate subgroups of r which are not disjoint. I

REMARK 3. Just as was done for loops in Proposition 2, we can deriv_e necessary and sufficient conditions on the current graph K in-order that K be simplicial. However, there are several ways in which K may inherit multiple edges from K, so we will not present the details. The arguments are similar to those in the proofs of Propositions 2 and 4.

Corollary 1. Suppose that K is simplicial with currents in a group r of order 2, and that for each edge e of K the intitial excess current and the terming excess current of e generate disjoint subgroups each of order n in r. Then K is isomorphic (as an undirected graph) to the composition of the (undriected) graph K over n independent vertices.

Proo$ By “undirected graph” we mean that which is obtained by suppressing the directions in an oriented 1 -cell complex. Since K is simplicial, by Proposition 4 we get is simplicial. By Proposition 3, each vertex v of valencyp in K lifts to n2/n = n vertices of valency np in K,Each edge e kom u to v in K lifts to n2 edges from w-l(u) to W ’ ( v ) in K, and since K is simplicial, this means that each-edge e of K lifts to a copy of the complete bipartite graph&,. Therefore K = K[nK,J (see [7, p. 221 for the definition of the composition G[HI). I

To illustrate an application of Corollary 1, we shall compute the orientable genus of the graph K2,,n,n,n for all odd integers n.

Example 3. Consider the triangular embedding (Fig. 18) of K,,,,,,, in the sphere, with currents in the additive group 2, X 2,. Since 2, X 2, is abelian, the initial excess current of an edge e from edges out of ZI to a vertex w is just the sum ~ ( v ) of the currents of the oriented edges out of I/, and the terminal excess current of e is just the sum ~ ( w ) of the currents ofthe oriented edges out of w. We compute that &(a) = (2,2), &(a’) =

= (-2,0), and &(d) = (0,-1). If n is odd, then each of these elements is of order n in 2, X Z,, and except for the pair a, a’, for any two distinct vertices v,w we have that E(v) and &(w) generate disjoint subgroups of 2, X 2,; but there is no edge of a to a’, so this current graph K = K2,1,1,1~atisfies the hypothesis of Corollary 1. By Corollary 1 and Proposit@ 3, K = K,,,,,,,, has a triangular-embedding in an orientable surface S, and the Euler characteristic of S is 5n - 9n2 + 6n2 = n(5 - 3n) = 2 - 2g. Thus the genus g of K2,,,,,,, is (3n - 2)(n - 1)/2 for all odd n. Other triangular embeddings ofK2,,,,,,, and . .,, (where i is the number of partite sets) appear in Jackson [8].

( l , l ) , &(b) = (2,1)+(-2,-2)+(-1,-1) = (-1,-2), &(c) = (O,l)+(-2,-1)


FIGURE 18.

As a final application of our theory, we prove a generalization of Example 3 and of a theorem of [lo].

Theorem 3 . Let k L 3 be an integer. Then there exists a positive integer Mk with the following property: for every positive integer n relatively prime to k f k and for every simplicia1 undirected graph G having chromatic number k and having a triangular embedding in some orientable surface, the composition G[nKl] also has a triangular embedding in an orientable surface.

boo$ Let Y be the set of all vectors ( E ~ , E ~ , - . * , E ~ ) such that each E,.€ {O,* 1 ,+2) and at most k-2 of the components ei are zero. List the vectors in Y a s y l , y 2 , - - * y m , where m = I YI = 5 k - 4k - 1. Let nj = Iyj *

(1 , 1, - - * ,l )I be the absolute value of the sum of the components ofyj, and let N be a positive integer such that N > 2 + 2k. Then for a l l j = 1, - - - , rn we haveyj-(1JVJV2,. - *,Nk- l ) # 0 andyj-(1JV,N2; * . ,Nk-l) # f N i n . J for i = 0,1, - * * , k - 1. (This is because every non-negative integer has a unique N-adic expansion of the form co + clN + c 2 f l + * * - + c,N, where each cie (0,1, - * a , N - l}.) Let p 1 ,p2, * * - , pr be all the distinct primes occurring as divisors of the (nonzero) integers yj (1 , N , p , * - ,Nk- ' ) and yj - (l,N,@;**,N"-') i N'nj f o r j = l , . . . , m and i = 0,1,-*., k - 1. Let i b fk = p1p2 * - - p a , and let n be a positive integer relatively prime to k f k .

Note that 2 occurs among thep, since 2 divides fl + N = (O,l,l,O, * . ,0) - (1 ,N,

Suppose G is a simplicia1 undirected graph, of chromatic number k, which has a triangular embedding in an orientable surface S. Consider G as embedded triangularly in S, and color the vertices of G with k colors

- ,Nk-'), so n is odd.


0,1, - * , k - 1 so that the independent sets Ai(i = 0,l , - - - ,k - 1) of vertices colored i have the property that lAol is as large as possible. Then each vertex inAi(i > 0) is adjacent to some vertex of the set A,, and each vertex v of & is adjacent to at least two other vertices in different sets A,,Aj (0 < i < j ) , since v is in some triangle of G.

If e is an edge of G with endpoints u,v where u E A,, v E Aj and i < j , assign the preferred direction to e from u to v ; and if 0 < i < j , then give e the current (0,O) in Z,, X Z,.

Consider the bipartite subgraph of G spanned by the vertex set Ai(i > 0) and the subset Bi of A,,, where Bi consists of all vertices of A. which are adjacent to some vertex of A,. By a slight modification of Lemma 2 in [ 101, we may assign currents of the form (O,O),+( 1 ,Ni), and f 2 ( l,Ni) to the edges from Bi to Ai, in such a manner that the initial excess current of each such edge is either (1,") or (-1,-N'), and the terminal excess current of each such edge is one of (l,W'),(-l,-F), (2,2N') or (-2,-2N'). We assign currents to the edges from A, to each Ai( i > 0) in this way.

Now each edge of G has been assigned a current in 2, X Z,, and we claim that the initial and terminal excess currents of each edge e are of order n in Z, X 2, and generate disjoint subgroups of Z, X Z,,. To see this, we must consider two cases.

Case 1. Edge e is from Ai to Aj , where 0 < i < j . Then the initial excess current of e has the form (Si, SiN'), where 8, E {+1,+2} and the terminal excess current of e has the form ( S j , SjNj), where Sj E { f 1, f 2). Ift( Si,6,N') = (0,O) in 2, X Z, then tSi O(mod n), but Si is a unit mod n, since n is odd; therefore t = 0 (mod n) and (Si,SiN') has order n in 2, X Z,. Similarly, so does (Sj,SjNJ>. Next, suppose (0,O) # s(S,,S,N') = t(Si,SjNj) for some integers s,t . Then (s6, - tSj,sGiNi - tSjNj) = (0,O) in 2, X Z,, so sSi = t S j (mod n). Thus 0 = sSi(N' - Nj) (mod n). But" - Nj = y, - (1 ,N,A@, - * * ,Nk-'), where y, E Y is that vector with ith component 1, j th component - 1, and 0 components else- where. Thus N' - P' is relatively prime to n, so we get 0 = sSi (mod n), which implies (0,O) = s(Gi,SiN'), contrary to hypothesis. Therefore ( a;, 8,N') and ( S j , SjNj) generate disjoint subgroups of order n in Z, X 2,.

Case 2. Edge e is from A, to A, for some i > 0. Then the terminal excess current of e is of the form ( Si,S,Ni), where 6, E {f 1 , f 2 } , and the initial excess current ofe is of the form(? cr ,X E J ' ~ ' ) , where E, E (l,-l) and the sums run over r = 1,2; - -, k. kt yq = - - -, E ~ ) be the corresponding vector in Y. Then the initial excess current of e is (y;(l,l,. * a , l ) , y;(l,N; - , N k - ' ) ) , and since the second of these


components is an integer relatively prime to n, this element of 2, X 2, has order n.

Now suppose (0,O) # s( ai,6,N') = t( x e r , x E , W - ' ) in 2, X 2, for some integers s,t. Then sdi = tv, - ( l , l , * * * ,1) (mod n ) and s6," = tys (1 ,N, - * , N k - ' ) (mod n). Therefore tys - [( 1 , N , * - - , N k - 1) - M ' ( l , l , * * * , l ) ] = 0 (modn). Buty;[(l,Nk-') - N ' ( l , l , - - - , l ) ] is an integer relatively prime to n (by the construction of Mk), thus t = 0 (modn).Then (0,O) = t ( z ~ , , x &,&,Nr-') inZ, X Z , , contrary to hypothesis.

It follows that, for every edge e of G, the initial and terminal excess currents of e generate disjoint, order n subgroups of 2, X 2,. By Corollary 1 and Proposition 3& i;' = G [ n , K , ] has a triangular embedding in an orientable surface S , concluding the proof of Theorem 3.

We note that Theorem 3 determines the orientable genus of a remarkable variety of graphs, namely infinitely many G [ n K l ] for G any triangulation of any orientable surface. The proofs are constructive: we first compute the chromatic number k = x(G), then we compute Mk, and we take any n such that (n,Mk) = 1.

In [lo], we obtained the weaker theorem for the cases k = 3 and k = 4, where one can take M3 = 1 and M4 = 30 in the statement of the theorem.

We remark that the theorem has variations in which G may be arbitrarily embedded in an orientable surface S , and then-for ( n , k f k ) = 1, G[nkl] will Lave an embedding in some orientable surface S with the faces of G[nK,] in S having the same sizes of those of G in S.

r r

r r

I

References

[l] A. Bouchet, Triangular embeddings into surfaces of a join of equi- cardinal independent sets following an Eulerian graph. Lecture Notes in Math. #642. Springer-Verlag, New York (1976) 86-1 15. RH. Fox, Covering spaces with singularities. Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz. Princeton Uni- versity Press, Princeton, NJ ( 1 957). J.L. Gross, Voltage graphs. Discrete Math. 9 (1974) 239-246. J.L. Gross and S.R Alpert, The topological theory of current graphs. J. Combinatorial Theory, Ser B 17 (1974) 218-233. J.L. Gross and T.W. Tucker, Quotients of complete graphs: Revisiting the Heawood mapcoloring problem. PaciJic J. Math. 5 5 (1974) 391- 402. I

J.L. Gross and T.W. Tucker, Generating all graph coverings by permutation voltage assignments. Discrete Math. 18 (1977) 273-283. F. Harary, Graph Theory. Addison-Wesley, Reading, MA (1969).

[2]

[3] [4]

[5]

[6]

[7]


[8] B. Jackson, Triangular embeddings of K((i-2)n,n,* - -, n) . J; Graph Theory 4 (1980) 21-31.

[9] B. Jackson, Generative m-diagrams and embeddings of K&. To appear.

[lo] T.D. Parsons, T. Pisanski, and B. Jackson, Dual embeddings and wrapped quasicoverings of graphs. Discrete Math. 31 (1980) 43-52.

[ 111 G. Ringel, Map Color Theorem. Springer-Verlag, New York (1974). [ 121 G. Ringel and J.W.T. Youngs, Solution of the Heawood mapcoloring

problem. Proc. Natl. Acad. Sci USA 60 (1968) 438-455. [ 141 S. Stahl, Generalized embedding schemes, J. Graph Theory 2 (1978)

[15] AT. White, Graphs, Groups, and Sutjiaces. North-Holland, Am- sterdam (1973).

[16] A.T. White, Graphs of groups on surfaces. Combinatorial Surveys: Proceedings of the Sixth British Combinatorial Conference. Aca- demic Press, London (1977) 165-1 97.

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A duality theorem for graph embeddings

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