embeddings of bipartite graphs

Embeddings of Bipartite

Mohammed Abu-Sbeih Graphs

and T. D. Parsons THE PENNSYLVANIA STATE UNIVERSlTY

UNIVERSITY PARK, PENNSYLVANIA

ABSTRACT

If G is a bipartite graph with bipartition A, B then let G,,,(A, B) be obtained from G by replacing each vertex a of A by an independent set a,, ..., a,, each vertex b of B by an independent set 61, ..., b,, and each edge ab of G by the complete bipartite graph with edges a,b, (1 =s i s m and 1 =s j s n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of G,,,(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of ”excess-current graphs.” They include new genus embeddings for a large class of bipartite graphs.

1. INTRODUCTION

Let G be a graph with vertex set V and edge set E. If k: V + { I , 2, 3, ...} is a function mapping vertex u to the integer k,, then let G(k) denote the graph with vertex set {uj 1 u E V and 1 S i S k,} and with the edge set {up j I uu E E , 1 6 i s k,,, and 1 6 j s k,}. In the special case where G is bipartite with bipartition A , B and k(A) = {m} and k(B) = {n}, then G(k) is also bipartite and is called G,,,(A, B). Usually G,,,(A, B) is not isomorphic to G,JA, B).

Note that when k ( V ) = { n } . the graph G(k) is just the composition G[K,; , ] . For this reason, we call G(k) a “generalized composition” of G over the various independent sets of size k,. When G is bipartite and m = n , then G,,,(A, B ) = G,,,(A, B ) = G[K,I.

Whenever there is an integer N such that for all uu E E we have k , k, = N, the graph G(k) is a special type of “wrapped covering” (formerly, a wrapped quasicovering) in the sense of Jackson, Parsons, and Pisanski [5, 61. This condition is easily seen to imply that, for any connected G , either k(V) = {n} where nz = N or else k ( V ) = {m, n } and G is bipartite with bipartition A , B where A = k - ’ ( r n ) and B =

Journal of Graph Theory, Vol. 7 (1982) 325-334 Q 1983 by John Wiley & Sons, Inc. CCC 0364-9024/83/030325-10$02.00

326 JOURNAL OF GRAPH THEORY

k - ' ( n ) . In these cases, we might hope to apply the theory of wrapped coverings to lift embeddings of G in surfaces S to embeddings of G(k) in surfaces 3, such that each face f of G in S lifts to faces of the same size as f i n the embedding of G(k) in 3; such a lifting of embeddings will be called afuce-lift of G to G(k). For a precise account of the way in which faces of G in S correspond to those of G(k) in 3, and for details about wrapped coverings, we refer the reader to the papers [5 ,6 ] . However, it will not be necessary to know such details in order to read the present paper.

We shall concentrate here on face-lifts of bipartite graphs G to the graphs Gm,n. To simplify notation, henceforth G will always denote a connected bipartite graph with a fixed ordering A, B of its bipartition, and Gms,,(A, B) will be abbreviated to Gm.n. We observe that (Gm,n)r,, = G,,,,, holds if we adopt the convention that the ordering of the bipartition A', B' of G,,,(A, B ) is inherited in the obvious way from that of the bipartition A, B of G.

All graphs will have no loops or multiple edges, and all embeddings will be cellular embeddings in orientable surfaces.

Our starting point will be the following proposition, which is an immediate consequence of Propositions 3 and 4 of [ 5 ] :

Proposition 1. Let B be a connected bipartite current graph with no multiple edges and with currents in a finite Abelian group r of order mn. For each vertex u of G, let E ( U ) be the sum of all outward-directed currents at u, and let k, be the index in r of the subgroup ( E ( u ) ) generated by E(u) . Suppose that G has the bipartition A, B, that k, = m for every u E A, kb = n for every b E B, and that for each edge ab of G we have that ( ~ ( u ) ) rl ( ~ ( b ) ) is the trivial subgroup of I'. Then every cellular orientable embedding of G face-lifts to a cellular orientable embedding of G m . n *

2. FACE-LIFTS OF BIPARTITE GRAPHS

Let F be a family of graphs. An F-factor of a graph G is a spanning subgraph each component of which belongs to F.

Theorem 1. Let G be a connected bipartite graph, and let m and n be positive integers. Suppose that either

or (1) G has a {P,}-factor,

(2) both m and n are odd and G has a {P.,, P,}-factor. Then every orientable embedding of G face-lifts to an orientable

embedding of G,,,n.

EMBEDDINGS OF BIPARTITE GRAPHS 327

Proof. Let Z,, denote the additive cyclic group of order n (if 12 = 1 , then Z , = (0) is the trivial group). Consider the two figures below, which give current graphs with currents in the group Z,, x Z,,, for the paths P4 and Pg. Let A , B be the bipartition of G .

In Fig. 1 we compute that & ( a , ) = ( - 1 , 0), ~ ( b , ) = ( - I , I ) - ( - 1 , 0) = (0, l), &(aZ) = (0, 1) - ( - 1 , 1) = ( I , 0), and &(bZ) = -(O, 1) = (0, - 1).

Suppose that G has a (P,}-factor, that is, a spanning forest of paths with four vertices. Each such path can be oriented as in Fig. 1, so as to go from a vertex in A to a vertex in B , and we assign currents in Z,, x Z , to the oriented edges of each such path as shown in Fig. 1 . AU remaining edges (not in the chosen spanning forest of P4’s) are oriented arbitrarily and are assigned the current (0. 0). We now have for each a E A that &(a) is either ( I , 0) or ( - 1, 0) , while for each b E B , ~ ( h ) is either (0, 1) or (0, - I ) . Therefore for all a E A , (&(a)) = Z,, x (0) has index m in Z,, x Z,,,; for all b E B , ( ~ ( b ) ) = (0) x Zm has index n ; and for each edge ab of G, (&(a)) fl ( ~ ( b ) ) = ((0, 0)). It follows from Proposition 1 that every orientable embedding of G face-lifts to one of G,,.,, (for all m, n ) in this case.

Next, suppose that m and n are both odd and that G has a (P4, P6)-

factor. Using both Figs. 1 and 2, we may similarly orient the edges of G and assign currents to them in Z,, x Zm in such a way that for each u E A , we have &(a) is one of ? ( I , 0) or +(2, O), for each b E B , we have ~ ( b ) is one of k(0, 1) or +(O, 2). In this case, once again (&(a)) has index m, ( ~ ( h ) ) has index n, and ( ~ ( u ) ) r l ( ~ ( h ) ) = ((0, O ) } , so the conclusion follows from Proposition 1. I

Corollary 1. Suppose that G is a connected bipartite graph with a 2- factor. If m and n are both odd, then every orientable embedding of G face-lifts to G,,,, .

Proof. Each component of a 2-factor of G is a cycle of even length 4 or more. A cycle of length divisible by 4 has a {P,)-factor; every other cycle of even length has a (Pj, Pg}-factor using just one P b . Therefore G has a { P 4 , P,}-factor, and the result follows from Theorem I . I

Remark 1. A regular connected bipartite graph on more than two vertices contains two disjoint perfect matchings, and so has a 2-factor. This follows from a well-known theorem of Hall [4]; also see Exercise 5.2.3


of the text by Bondy and Murty [2]. Therefore Corollary 1 applies to regular graphs G (except K 2 ) .

Corollary 2. Suppose that G is a connected bipartite graph with a 1- factor. Then every orientable embedding of G2,z face-lifts to Gz,,z, for all positive integers r , s.

Proof. The edges of a 1-factor H of G correspond to the components of a {C4}-factor of G2,z. Therefore G2,* has a {P,}-factor. By Theorem 1, each embedding of G2,2 face-lifts to (G2.2)r,s for all r , s and (G2,2)r,s is isomorphic to G2r,2s. I

We shall now apply these results to obtain genus embeddings for various bipartite graphs. First, we note a very strong theorem of White [lo] which implies that for every connected graph G with at least two vertices, the composition graph GLX,, J has a quadrilateral embedding for every even n. As was noted previously, when G is bipartite and m = n then Gm,n = G[K,,]. Thus Theorem 1 and its corollaries will allow us to generalize White’s result, for the special case of bipartite graphs G, in two directions: first, when m = n and n is odd, we will get quadrilateral embeddings of G[K,,]; and second, when m # n , we will get quadrilateral embeddings of the “generalized compositions” Gm,n.

Theorem 2. Let G be a connected bipartite graph. Then Gm,n has a quadrilateral embedding in each of the following cases:

( I ) m and n are both even, (2) G has a quadrilateral embedding and a {P4}-factor; (3) G has a quadrilateral embedding, a { P 4 , P,}-factor, and both m and

(4) G has a quadrilateral embedding, a 2-factor, and both m and n are n are odd;

odd.

Proof. Cases (2) and (3) follow directly from Theorem 1. Case (4) is implied by case (3) (see Corollary I) , and is just stated for convenience. We now prove case (1). Suppose both m, n are even and let m = 2r and n = 2s. By White’s theorem, G2r,zs = Gr~, [K2] has a quadrilateral embedding. I


Remark 2. Let G be a connected bipartite graph with parts of size a, p (a 5 p) and with q edges. If Gm,n has a quadrilateral embedding, then by Euler’s formula we get the (orientable) genus Y(G,, ,~) = 1 + f[qmn - 2(am + pn)]. Example 1. Let R be any connected regular bipartite graph, and let G = Rk be the k-fold Cartesian product R x ... x R , for any k a 2. By a result of Pisanski [7], C has a quadrilateral embedding. By using a perfect matching in the last factor of the product Rk, it is easy to see that G has a {C4}-factor, and hence a {P,}-factor. Therefore by (2) of Theorem 2, Gm,n has a quadrilateral embedding for all positive integers m and n. Using the Remark 2, we can easily compute the genus of Gn,.n. For example, when R = K2 so that G is the k-cube Q k , we obtain the genus -y[(Qk),,,] = 1 + 2k-3[kmn - 2(m + n)] for all m , n.

Example 2. Let G be minus a I-factor. It is easy to find a quadrilateral embedding of G in the torus. Since G has 10 vertices, it cannot have a {P,)-factor, but it obviously has a {P,, P,}-factor. If Gm.n has a quadrilateral embedding then Remark 2 shows that m + n must be even (since a = 5 = p and q = 20 for this G). By cases (1) and ( 3 ) of Theorem 2 the graph G,,, does have a quadrilateral embedding when m + n is even, but as we have seen, it has no such embedding when m + n is odd. Therefore the quadrilateral embedding of G cannot face-lift to Gmsn when m + n is odd, so that some restrictions are indeed necessary on m and n in Theorem 1(2), where the graph has a {P , , P,}-factor but no {P4}- fact or.

Bipartite graphs with F-factors, for any family F of paths having an even number of vertices, must have bipartitions in which A , B have equal cardinalities: a = p. Therefore, to obtain a theorem similar to Theorem 1 for graphs with a # p , we need some current graphs for paths with an odd number of vertices. Figures 3 and 4 below again have currents in Z,, x Z,:

The current graph of Fig. 3 has ~ ( b , ) = (0, 21, &(al) = (1 , 01, ~ ( 6 ~ ) = (0, - I ) , &(aZ) = ( - 1, 0) , and 4 b 3 ) = (0, - 1). The current graph of Fig. 4 has ~ ( b , ) = (0, I ) , &(a, ) = (2 , O ) , &(h2) = (0, 1) &(a2) = ( - I , 01, &(b3) = (0, - I ) , &(a3) = ( - 1, O ) , and 4 h 4 ) = (0, - 1 ) . One obtains two more such diagrams by interchanging everywhere a and h (so a , becomes b l , h2 becomes a 2 , etc.) and by replacing each current (x, y ) on an arc by the current ( y , x). This gives two diagrams for P5 and two


for P7 in which every &(ai) is one of k(1, 0), r ( 2 , 0) and every ~ ( b , ) is one of k(0, l ) , k(0, 2). Using the diagrams of Figs. 1 and 2 along with these new ones, the same sort of argument which proved Theorem 1 now proves

Theorem 3. Let G be a connected bipartite graph with a spanning forest of paths of length 3 or more. Let m and n both be odd. Then every orientable embedding of G face-lifts to G",.,. In particular, if G has a quadrilateral embedding then so does G,,,,. I

We remark that we need only use paths PL for 4 S k 5 7 in Theorem 3 , because every P, for r 2 8 has a {P4, P5, P, , P7}-factor. Also we note that slightly stronger results may be true in special cases, using only the above figures; for example, we see that Fig. 3 actually works whenever m is odd and n is arbitrary, so that if G has a {P4, P,}-factor in which every P5 starts from a vertex in B , then embeddings of G face- lift to G,,,., for all m, n such that m is odd. Similar slight improvements involve uses of P4, P 5 , P7 in appropriate combinations.

Up to now we have used F-factors only for F some family of paths. We did this because paths are so simple, and they already yielded what we felt were some nice theorems. However, the use of current graphs is somewhat of an art, and the possibilities are not exhausted by the use of paths alone. Consider Fig. 5 below, where the currents are in 2, for an odd integer r , and where n is odd and n 2 3:

Here the current graph is Kz,,,. We compute that = 0 = &(aZ) and that ~ ( b , ) = - 1, E ( & ~ ) = - 1, &(b3) = 2, and (if n > 3 ) that ~ ( b , ) = 1 or - 1 for each i > 3 . It follows from Proposition 1 that every orientable embedding of K2,,, face-lifts to (K2, f l ) r , l = K2.,. In particular, the quadrilateral embedding in the sphere (shown in Fig. 5 ) lifts to a quadrilateral embedding of K2,., whenever r and n are odd.

Ringel [91 showed that y(K,~,) = [b(m - 2)(n - 2)], and that an orientable quadrilateral embedding exists for K,." when either both m and n are even or when one is odd and the other is congruent to 2 modulo &that is, precisely when Euler's formula allows such an embedding. The latter embeddings are provided by our Fig. 5 (along with the symmetry interchanging the roles of m and n ) . The former embeddings of K,,,, for m, n both even are provided by our Theorem 1, since K2.2 has a {P4}- factor and a quadrilateral embedding, so that (K2.2),.s = K2r,2s has a quadrilateral embedding for all r , s . (Alternatively, the existence of a


quadrilateral embedding for KZr,2s follows from White’s theorem, since K2r.2s = Kr. , [E2] . ) We include this example to show that the technique of “excess-current graphs” can be used to advantage even when the bipartite graph G has no F-factor for F a family of paths of length 3 or more, because for n > 3, clearly K 2 , , has no such F-factor.

When m = n and n is odd, we can obtain face-lifts of G to G,,n = C[x , ] without any hypothesis about F-factors for G.

Theorem 4. Let G be a connected bipartite graph with at least three vertices, and let n be odd. Then every orientable embedding of G face- lifts to G[K,,].

Pruof. Let T be a spanning tree of G. We shall show how to orient the edges of T and assign these edges currents in Z , x Z , so that for each vertex a E A in T there is an excess current &(a) in the set { & (1, I ) , k(2, l), & ( l , 2)) and for each vertex b E B in T there is an excess current ~ ( b ) in the set { -+ (1, 0), & (0, 1)). If we are successful in doing this, then we may arbitrarily orient each edge of G not in T , and assign it the current (0, 0); the theorem will then follow from Proposition 1 .

First, suppose that T is a star of degree d with central vertex u. Figure 6 shows an appropriate current distribution for T according to the parity of d and to whether u is in A or in B .

Now we shall proceed by mathematical induction on the number of edges of T. We may assume that T is not a star, so that T has some


u € A, d even

?

u C B, d even

t

u € A , d odd u € B, d odd

FIGURE 6.

edge ab (where a E A and b E B) whose deletion from T yields two subtrees, T I (containing a) and T2 (containing b), neither of which consists just of a single vertex. Let ui be the number of vertices of Ti (i = 1 , 2); then if uj > 2, we may apply the hypothesis of induction to Ti to obtain a “good” current distribution for it . If both u , , u2 > 2, then these currents for TI, T2 plus the assignment of current (0, 0) to ab (oriented arbitrarily) give good currents for T. If u, = u2 = 2, then T is a P4 of the form b,aba,; here orienting this path from b, to a , and assigning the currents ( - 1 , 0), (0, I ) , (1, I ) to bla , ab, ba, works.

This leaves us with the cases u I = 2 < u2 and u , > 2 = u2. In the former case, there are good currents for T2 such that ~ ( h ) = (1, 0); and in the latter case there are good currents for TI such that &(a) is either ( I , 1) or (2, I ) . [To see this, note that we may multiply all currents by - 1 and/or replace each current (x, y ) by (y, x), so as to change all excess currents (w, z ) to one of ( - w , - z ) , (z, w), or (-2, - w ) . ] For these final two cases, see Fig. 7.

In part (i) of Fig. 7, TI consists of the single edge ab’; the currents of T2 are good and make ~ ( b ) = ( I , 0) in T 2 , but the currents shown for ab‘ and ab now make ~ ( b ) = (0, - 1 ) in T, and &(a) = (2, I ) , ~ ( b ‘ ) = ( - 1 , 0).

In parts (ii) and (iii) of Fig. 7, T2 consists of the single edge a’b, and


the current shown, along with the appropriate currents for T I , give good currents for T . This completes the proof of Theorem 4. I

Theorem 5. Let G be a connected bipartite graph with at least three vertices. If G has an orientable quadrilateral embedding, then so does the composition G[xn] for all positive integers n.

Proof. When n is even, this follows from White’s theorem (and the hypothesis of a quadrilateral embedding for G may even be dropped entirely). When n is odd, this follows from Theorem 4. I

Andre Bouchet has informed us that he has obtained a (not yet published) result stronger than our Theorem 4: if G is any graph with even faces in a surface S, then this embedding face-lifts to one of G[K,] in a surface

of the same orientability characteristic as S , for all odd n.

Remark 3. Since most applications of the method of “excess-current graphs” require use of some sort of F-factor of the base graphs G (as in most theorems above), it is worth noting that

(1) if G has an F-factor, then so does every graph which covers G,

(2) if G has an F,-factor and F , C F 2 , then G has an F,-factor. and


Finally we remark that although we have mostly been concerned with quadrilateral embeddings of the graphs Gm,n (since they are genus embeddings when they exist), it should not be overlooked that Theorems 1 and 3 give many other potentially interesting embeddings of the graphs Gm.n. For example, the dual graph G of graph K7 triangularly embedded in the torus is a cubic bipartite graph with 14 vertices and 7 hexagonal regions; by Remark 1 , the corresponding graphs Gm,n have hexagonal embeddings whenever both m and n are odd.

References

[ l l L. W. Beineke and F. Harary, The genus of the n-cube. Canad. J. Math. 17 (1965) 494-496.

[2] J . A. Bondy and U. S. R. Murty, Gruph Theory with Applications. North-Holland, New York (1977).

[3] J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltage assignments. Discrete Math. 18 (1977) 273-283.

[4] P. Hall, On representatives of subsets, J. London Math. Soc. 10

[5] B. Jackson, T. D. Parsons, and T. Pisanski, A duality theorem for graph embeddings. J. Gruph Theory. 5 (1981), 55-77.

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

[7] T. Pisanski, Genus of Cartesian products of regular bipartite graphs. J . Graph Theory. 4 (1980) 31-42.

[8] G . Ringel, Uber drei kombinatorische Probleme am n-dimensionalen Wurfel und Wurfelgitter. Abh. Math. Sem. Univ. Hamburg. 20 (1955)

[9] G. Ringel, Das Geschlecht des vollstandigen paaren Graphen, Abh.

[lo] A. T. White, On the genus of the composition of two graphs, PuciJic

(1935) 26-30.

10-19.

Math. Sem. Univ. Hamburg. 38 (1965) 139-150.

J . Math. 41 (1972) 275-279.

embeddings of bipartite graphs

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