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Embeddings of Cartesian Products of Nearly Bipartite Graphs

- - Bojan Mohar

Tomai Pisanski UNIVERSITY OF LJUBLJANA

SLOVENIJA, YUGOSLAVIA

Arthur T. White WESTERN MICHIGAN UNIVERSITY

KALAMAZOO, MICHIGAN, USA

ABSTRACT

Surgical techniques are often effective in constructing genus embeddings of Cartesian products of bipartite graphs. In this paper we present a general construction that is "close" to a genus embedding for Cartesian products, where each factor is "close" to being bipartite. In specializing this to repeated Cartesian products of odd cycles, we are able to obtain asymptotic results in connection with the genus parameter for finite abelian groups.

1. INTRODUCTION

Surgical techniques are often relevant for constructing efficient orientable embeddings of graphical products, particularly for Cartesian products where each factor is a bipartite graph; see, for example, [8]. By the Fundamental Theorem for Finite Abelian Groups, each such group has as a Cayley graph a repeated Cartesian product of cycles. Thus the genus parameter for finite abelian groups (the minimum genus among all closed orientable 2-manifolds in which some Cayley graph for the group can be embedded) is intimately connected with the genus of repeated Cartesian products of cycles.

In particular, if all the factors are even (and hence bipartite), then the construction of [8] applies; see [9]. If even one factor is an odd cycle (and hence not bipartite), the surgical construction of [8] does not apply. However, in many of these cases, current graph constructions of genus embeddings are possible; see [4]. In fact, a quadrilateral genus embedding was constructed when- ever such is compatible with euler considerations, unless at least one factor is a Journal of Graph Theory, Vol. 14, No. 3, 301-310 (1990) 0 1990 by John Wiley & Sons, Inc. CCC 0364-9024/90/030301-10$4.00

302 JOURNAL OF GRAPH THEORY

3-cycle or there are exactly three factors in all. Thus the graph C, X C, X C3 (and the corresponding group Z , X Z , X Z3) is particularly resistant to all the constructions mentioned above. In [ 5 ] it was shown that both this graph and its corresponding group have genus at most seven, and in [ l ] it was shown that both have genus at least seven. The construction of [ 5 ] was neither surgical nor current graph (nor of the dual voltage graph form), but was ad hoc: finding a rotation scheme so as to force the desired embedding.

In this paper we present a general construction that is close to being minimal for Cartesian products, if each factor is close to being bipartite. Essentially, it is the best possible extension of the construction of [8] to nonbipartite factors. In specializing this construction to repeated Cartesian products of odd cycles, we are able to obtain asymptotic results in many cases of interest, in relevance to the open cases of the calculation of the genus parameter for finite abelian groups.

We remark that, in the nonorientable case, if one factor in a repeated cartesian product is not bipartite, then the surgical construction is not affected (the handles added in the last stage of the construction can be “twisted” in 4-space to join two embeddings of all previous factors having the same orientation; this can not be done as efficiently in 3-space, which we regard as ambient for orientable embeddings). For nonorientable surgical constructions of Cartesian products, see [6 ] .

2. THE GENERAL CONSTRUCTION

Let H, K, and L be graphs. By H = K @ L we denote the factorization of H into K and L. This means that V ( H ) = V ( K ) = V(L) , E ( K ) f l E(L) = 4, and E ( K ) U E(L) = E ( H ) .

Now let G be a connected graph, with S a surface (a closed orientable 2-manifold) and 4: G + S a 2-cell embedding; then 4 is said to s-anticolorable if the set R of regions of the embedding can be partitioned into s + 1 classes R,,R,, . . . ,R,-where R, may be empty and, for each i (1 5 i I s), the regions in R, meet all the vertices of G. Thus we partially color the regions with s colors so that each vertex is touched by a region of each color. Then R, is the set of uncolored regions. The partition of R into Ro,R,, . . . ,R, is called an s-anticoloring of 4. It is immediate that s 5 6(g), the minimum degree in G.

Our goal is to construct an imbedding of the connected Cartesian product H X G commencing with an s-anricoloring of 4: G + S (for 0 5 i 5 s, let r, = IR,I) and a decomposition H = K El L such that K is bipartite, and as large as possible (and hence connected). Then the maximum degree and edge chro- matic number of K agree (see, for example, p. 228 of [2]), and we want A ( K ) = x ’ ( K ) 5 s. Let E ( K ) = U:=, E, be an s-edge-coloring of K, with Eo = E(L) and el = IE,l, 0 5 i 5 s .

The embedding of H X G is constructed in two stages. We start with a surgical embedding of K X G, in the usual manner facilitated by the bipartite nature of K. Then we add tubes to accommodate the remaining IE(L)I IV(G)( = e,,(V(G)I edges of H x G.

GRAPH EMBEDDINGS 303

Stage 1

For each vertex u E V ( K ) we take a copy G, of G embedded into a surface S,. The surfaces S, are all homeomorphic (to, say, S) and are mutually disjoint and exterior to one another. Let V ( K ) = K , U K2 be a bipartition (so that each edge in K joins a vertex in K, to a vertex in K 2 ) . Then we always choose the same embedding 9: G, + S,, except that if u E K , we choose the clockwise orientation for S,, whereas if u E K2 we choose the counterclockwise orientation. For each i (1 5 i I s) and for each edge e = { u l , u2} E Ei of K there are ri tubes added surgically (one tube between each region in Ri of G,, in S,, and the mir- ror image of that region in GUZ in S,> to carry the \V(G)l copies of e in K X G. The opposite orientations for S,, and Su2 ensure that each tube can accommodate its requisite edges without intersection.

In the embedding of K X G we therefore have:

(i) IV(K)( copies of 4: G --* S; (ii) ejri tubes, for each 1 5 i I s.

Let g be the genus of S, with g the genus of the surface s’ resulting from the stage 1 construction; 4: K x G +. 2. Let n = Jv(H)( = ( V ( K ) ] . Then

S

g = ng + C ei(ri - 1) + P ( K ) , i= I

where P ( K ) = IE(K)I - IV(K)I + 1 is the Betti number of K. (To verify ( l ) , it is helpful to count the “holes” in s’, rather than the handles.) Simplifying in ( l ) , we obtain:

S

g = 1 + n(g - I ) + C e i r i . i = I

In preparation for iterating this process, we form the region sets 8, as fol-

(i) all regions on the tubes associated with color i; there are IV(G)le, such

(ii) all regions from R, on G, in S,, for each u E V ( K ) incident with no edge

lows: For 1 I i I s, we put into k,:

regions;

colored i; there are (n - 2e,)r, such regions.

Thus

Finally, we let &, consist of all the remaining regions, in the embedding I$ of K X G into $ (of genus 2). We note that the factor IV(G>l in the first product giving f j in (2) above can be lowered by using, as far as possible, alternate regions on each tube associated with color i.


Stage 2

Necessarily, each edge e E E(L) joins two vertices in the same partite set of V ( K ) ; thus a tube between corresponding regions of the corresponding surfaces (of the same orientation) can carry at most two edges. Let M be a maximum matching in G, with rn = [MI and t = IV(G)l - 2m. For each edge uu of M, add a tube in such a way as to preserve orientability and embed on it the two edges u X e and u X e. Add an additional tube for each of the remaining t vertices.

Thus we add, for each e E E(L) , rn + t = IV(G)l - rn tukes; the genus g for s’ increases, therefore, by e,(lV(G)I - rn); the final surface S has genus

I

H = 1 + I v ( H ) I (g - 1) + C e,r, + e,(lV(G)I - rn). (3) , = I

We state this result formally:

Theorem 1. Let the connected graph G have a 2-cell embedding of genus g that is s-anticolorable into region classes R,, where lR,l = r, (0 5 i I s). Let the connected graph H be decomposed as H = K 43 L, where K is bipartite and E ( K ) = UrZl E, is an s-edge-coloring of K , with E, = E(L) and [Ell = e, (0 I i 5 s). Let M be a maximum matching in G, with rn = [MI. Then the Cartesian product H X G has a 2-cell embedding of genus g, where

I

g = 1 + IV(H)I (g - 1) + e,r, + e,(lV(G)I - m) . I , = I

Now we consider two special cases. If G has a 1-factor, then IV(G)l = 2m, and

If G has a near 1-factor, that is if IV(G)l = 2m + 1, then

S

g = 1 + lV(H)I(g - 1) + z e , r l + e,(lV(G)I + 1)/2. (3b) I = I

If we wish t9 iterate the entire process, to embed (say) H ’ x (H x G), then we determine R, (or order F,, 0 5 i 5 s) for the embedding of H XsG into s” and continue as above. The challenge is to select the region sets R, as efficiently as possible (that is, so as to minimize the F,, 1 5 i I s). Often, as mentioned above, it is desirable to use, as far as possible, alternate regions on the tubes added surgically to embed H x G. If the stage 2 construction can be ac- complished withoq altering the regions in R,, as will be the case in Section 3, then we can take R, = i?,.


3. ASYMPTOTIC RESULTS FOR ODD CYCLES

We have seen that the graph C , X C , X C , is of particular interest to topologi- cal graph theory. Thus it seems natural to consider the family G, = C2,+, x C2,+, X C2,+ , , in applying the construction of Section 2 . Not only does C2,+, have a near 1-factor, but also C,,,, is nearly bipartite. Moreover, the bipartite subgraph P2,+, , lacking only one edge of C2,+ , , is an increasingly closer ap- proximation to C,,,, as k gets large. Hence it is reasonable to expect an asymptotic result from the construction.

First we establish a lower bound for the genus y(G,). As we will allow k to increase without bound, we can assume initially that k 2 2 . (Moreover, we know already that y ( G , ) = 7.) Thus G, has no 3-cycles, so that the lower bound y(G,) 2 1 - p / 2 + 4/4 of Corollary 6-15 of [lo] applies, where G, has p vertices and 4 edges. Therefore

y(G,) 2 1 - ( 2 k + l),/2 + 3(2k + 1)3/4

= 1 + ( 2 k + 1),/4.

To establish an upper bound for y(G,), we wish to apply (3b) of Section 2 . Here we take G = C2,+, X C,,,, and 4: G -+ S as the usual square tessellation of the torus. Then H = C,,,, = P2,+, @ L2,+, = K C3 L , where L2,+, consists of 2k + 1 vertices and one edge. We have g = 1 and s = 2 , with e , = e, = k and e, = 1 . Moreover, we can select R, and R, so that r , = r, = k 2 + 2k . (See Figure 1 , for the case k = 3. The regions in R, are indicated by their preparation for surgery. The regions in R , can be obtained by translating those in R , one unit to the right (say). The figure extends easily for larger k, by adding two rows at a time at the top and two columns at a time at the left. (For k = 2, we remove the two top rows and the two left columns of Figure 1 .)

Then, by (3b),

S

y(G,) I g = 1 + IV(H)I(g - 1) + z e i r i + e,((V(G)I + 1 ) / 2

= 1 + 2k(k2 + 2k) + ( ( 2 k + 1), + 1 ) / 2 i= I

= 2k3 + 6 k 2 + 2k + 2 .

Combining the upper and lower bounds, we find that

2k3 + 3k2 + ( 3 / 2 ) k + 5/4 5 y(G,) 5 2k3 + 6 k 2 + 2k + 2 .

Now, dividing by 2k3 and taking the limit as k approaches infinity, we find that


FIGURE 1

The construction just employed compares favorably with an index one voltage graph construction giving an upper bound for y(G,) asymptotic to 8k3 and with the construction of [ 111, which combines voltage graphs and surgery to get an upper bound asymptotic to 4 k 3 . In fact, the construction of Section 2 can be iterated, as indicated in that section, to inductively establish that, for m L 3 and fixed and as k approaches infinity,

y(C2,+,)" = (m - 2)2"-'k"

For an alternative approach, let g ( m ) = g for (CZlr+,)", with r(m) = F, = f , , i = 1 , 2 ; set A = 2k + 1. Then the recurrence r ( 2 ) = k 2 + 2k,r(rn + 1) = ((A" + 1) /2)k + r(m) (for rn 2 2 ; see ( 2 ) , and the remarks following ( 2 ) , in Section 2 ) is solved by

r(rn) = ( k m / 2 ) + ((A" - 1)/4), for m 2 2 .


In turn, this is used, in conjunction with (3b), to produce the recurrence g(2 ) = 1, g(m + 1) = Ag(m) + ( ( k + 1)/2)Am + k2m + ( 1 - 5 k ) / 2 (for m 2 2), which is solved by

g(m) = ( (m - 2)/4)Am + (m/4)A"-' - (1/2)A"-2 + (1/2)((A"-2 - 1)/(A - 1)) T (m/4)A + (m/4) + 1 , for m 2 2

Then we have that

1 + ( (m - 2)/4)Am I y(C2k+l)m I ( (m - 2)/4)A" + O(Am-') ,

and, for k (and hence A) + 03, the asymptotic result is as claimed. We established the inductive anchor (the case m = 3) of this asymptotic con-

struction above. In so doing, we illustrated the increased efficiency of the new construction over previous constructions, in obtaining specific embeddings for graphs such as G, = (C2k+1)3. Then we extended this calculation to arbitrary m 2 3, by two related methods. (Both rely upon the construction of Section 2; but whereas one is within an induction framework, the other solves the recurrence relation directly.) But yet another method is readily available for asymptotic results.

For application to the calculation of the genus parameter for finite abelian groups (see Section 4 ) , we are concerned with the graphs

where m,Imi+' , 1 5 i 5 r - 1 (we set m, = l) , and n , r 2 3. If n is even, then all cycles are even, and we know from [8] that

On the other hand, if n is odd, then let mk be odd but mP+' be even (where k is uniquely determined, 1 I k 5 I ) . Then, since an odd cycle C, is homeomor- phically a subgraph of the even cycle Cm+', and this feature is preserved by Cartesian products,


Thus we see that, as n approaches infinity,

This includes the previous result y(c2k+1)3 = 2k3 and its generalization to rn 2 3 as a special case, and seems to be a more efficient way to obtain that and similar asymptotic results. But if we want an upper bound for fixed k, this “short” method is not as good. For example, it gives Y ( C ~ ~ + , ) ~ 5 2k3 + 6 k 2 + 6k + 3, in excess of the upper bound 2k3 + 6k2 + 2k + 2 obtained previ- ously. For the next iteration, m = 4, the difference between the two upper bounds increases to 8k3( 1 + o( 1)).

4. APPLICATION TO THE GROUP GENUS PARAMETER

If r is a finite abelian group, then (see, for example, [3]) we can write

r = z,, x - . - x z,, x znI ,

where n, I n,+ ,, 1 5 i 5 r - 1. Since the Cartesian product operation for graphs exactly models the direct product operation for groups (see, for example, Theorem 4-19 in [lo]), we have that the repeated Cartesian product of cycles C, x - * - x C,, x C,, is a Cayley graph GA(r) for r; thus y ( r ) 5 y(GA(r)).

Collecting the results of [ 11, (41. [5], [9], and Section 3 of this paper, we can now state:

Theorem 2. Let r = Z,, x * * x Zn2 x Znl, where n,1ni+,, 1 5 i 5 r - 1, and n > 1 (unless Irl = 1); let N ( T ) = 1 + ( r - 2 ) lrl/4.

(1) ([9]) If r = 1, then y(T) = 0. ( 2 ) ([9]) If r = 2 and n, = 2 , then y(T) = 0. (3) ([9]) If r = 2 and n, > 2 , then y(T) = 1. (4) ([l], [ 5 ] ) If r = 3 and n, = n2 = . n 3 = 3, then y ( r ) = 7. ( 5 ) ([9]) If r = 3 and n, 2 4 and even, with N ( T ) an integer, then

r r ) = W).


(6) ([91) If r 2 3 and n , = 2, then y(T) = N ( T ) - kl r l /8 , where k( 1 5 k 5 r is uniquely determined by n, = 2, n,,, > 2.

(7) ([4]) If r 2 4,n, 2 4, and N ( T ) is an integer, then y(r) = N ( Q . (8) (Section 3) If r 2 3 is fixed and n, + m, then y(r) = N ( T ) . I

Since there is a similar structure theorem for finite hamiltonian groups, similar asymptotic results for the orientable genus parameter can be obtained for these groups (see [7] and [lo] for the known exact results). For example, a major open case is for r = Q X Zmr X * - . X Zm2 X Z,,, where Q is the quaternions, r 2 6, all mi are odd, and as usual mi I mi+, (1 5 i 5 r - 1). We will let m , go to infinity, so we can assume that m, 2 5. Then we use Corol- lary 13h, Lemma 25 (if r is even), and Theorem 15 of [7] (as in Case 10 of the proof of Proposition 24; we set r, = Q X Zm, X - X Z,,) to find

so that

If r is o( 1, we s i result.

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?(r) = 1 + ( r - 2)1r l /4 .

it off C,, for Zm3 as well, and obtain the same asymptotic

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