Quadrilateral Embeddings of Bipartite Graphs

Ian Anderson UNIVERSITY OF GLASGOW

SCOTLAND

ABSTRACT

Current graphs and a theorem of White are used to show the existence of almost complete regular bipartite graphs with quadrilateral embeddings conjectured by Pisanski. Decompositions of K, and K,,” into graphs with quadrilateral embeddings are discussed, and some thickness results are obtained. Some new genus results are also obtained.

1. INTRODUCTION

Quadrilateral embeddings of various graphs can be constructed by the use of current graphs regular of degree four. A simple example, showing how to obtain a quadrilateral embedding of the complete graph on &, the orientable surface of genus 43, is given by White ([13], p. 118). We shall assume that the reader is familiar with the basic ideas of current graph methods as described in that book.

Let G, denote the pseudograph obtained from an ngon by duplicating every edge. In the usual notation,p = n and q = 2n, so that q - p = n. Thus (see Ringel [ 111, p. 20) no rotation scheme can induce just one circuit if n is even. However, if n is odd, there does exist such a rotation. The rotations for n = 3, 5 , and 7 are exhibited in Figures 1 (b), (c), and (d), respectively, while Figure l(a) shows how each vertex rotation is determined. An induction argument then shows that the obvious extensions to larger n do indeed induce one circuit. The induction step is carried out in exactly the same way as the rotation for G7 is obtained from that for G3 by “opening up” G3 at A and inserting the extra piece which lies between the two points A in G,.

A variety of quadrilateral embeddings can now be obtained by suitable assignments of currents to the edges. To start with an easy example, assign the currents 1, 2,. . . , 2n in Z4n+l, as illustrated for n = 5 in Figure 2(a), to obtain in the standard way a quadrilateral embedding of K4n+l, i.e., since n = 2r + 1, a quadrilateral embedding of &+5 on S8r2+5r+, for each r. We note that the Euler formula permits such an embedding for K,,, only when m _= 0 or 5 (mod 8), so exactly half of the possible embeddings are thus exhibited.

Journal of Graph Theory, Vol. 5 (1981) 355-361 o 1981 by John Wiley & Sons, Inc. CCC 0364-9024/81/040355-07$01 .OO

356 J O U R N A L OF GRAPH THEORY

F I G U R E 1. Current graphs and rotations.

2. REGULAR BIPARTITE GRAPHS

If, instead, we assign the currents 1,3,5,. . . , 4 n - 1 in ZSn as illustrated for n = 5 in Figure 2(b), we obtain a quadrilateral (and hence genus) embedding of K4n,4n , i.e., since n = 2r + 1, of K8r+4, 8r+4 on S(4r+,)2. Thus we have a new proof of the genus result y(K2k,2k) = ( k - 1f in the case of k = 2 (mod 4). As is well known, the genus ofKmp was found by Ringel [lo], and since then other authors (see for example [ 6 ] ) have given alternative complete or partial proofs. If, however, we interpret the currents in Figure 2(b) as being in Z,, instead Of Z40, we obtain a quadrilateral (and hence genus) embedding of the regular bipartite graph obtained from K21,21 by removing a 1-factor. The first line of its rotation scheme is

FIGURE 2. Currents in (a) Z2, , (b) in Z40.

QUADRILATERAL EMBEDDINGS OF BIPARTITE GRAPHS 357

0 . 1 5 31 3 3 37 35 9 13 23 39 4 1 2 5 29 27 17 19 15 11 7 3.

In general, if we use G(m, d ) to denote a connected regular bipartite graph of degree d on 2m vertices, we can, by this method, exhibit a graph G(m, m - 1) with a quadrilateral embedding whenever m = 5 (mod 8). Such an embedding can exist only when m = 0 or 1 (mod 4), and the existence of such graphs is reported by Alavi et al. [l]. It has been conjectured by Pisanski [9] that a G(m, m - r ) with a quadrilateral embedding exists whenever m - r 2 4 and 4 I m(m - r). The special case of m = 2k has been confirmed by Alpert and Gross in [2]. Now by assigning the currents 1, 3,. . . , 4n - 1 to G, as in Figure 2(b) and by interpreting them as being in ZsntZr, we immediately obtain a graph G(4n + t, 4n) with the required property for all odd n and all t 2 1. We thus exhibit, for each fixed r, a graph G(m, m - r ) as required for all m = 4 + r (mod 8), m 1 12 + r. In particular, the graph obtained for r = 2 is a complete bipartite graph with a Hamiltonian cycle removed.

The necessary condition of Pisanski above, that 4 I m(m - r), requires that mbeevenifr iseven;m=Oor 1 (mod4)ifr= 1 (mod4);andm=Oor3 (mod 4) if r = 3 (mod 4). So for each r we obtain a graph G(m, m - r ) by this method in precisely one-quarter of the possible cases. For even r, however, we can construct a suitable G(m, m - r ) in all possible cases. As in [ 121, define the composition G(H) of two graphs G, H to be the graph with vertex set V(G) X V(H), with vertices (u , v ) and (u’, v ’ ) joined by an edge precisely when either (i) u = u’ and vv’ is an edge of H, or (ii) uu’ is an edge of G. White [ 121 proved among other things that if G is a connected bipartite graph withp vertices and q edges, then

Take as G the complete bipartite graph K , , with h < n edge-disjoint 1 -factors removed. For example, we could take G to be the graph with vertices al, . . . , a,, b,, . . . , b, where ai is joined to bj except when j = i, i + 1,. . . , i + h - 1 (interpreted modulo n). Then p = 2n, q = n(n - h ) , G(R2) is a G(2n, 2n - 2h), and the genus of G(R2) is

n(n - h ) - 2n + 1.

Now, for a genus embedding of this graph on S, we have by the Euler formula

y > 1 -b+$.aq= 1 - 2 n + n ( n - h )

with equality if and only if the genus embedding is quadrilateral. The following is therefore now proved.

Theorem 2.1. There exists a regular bipartite graph of degree m - r on 2m vertices, with a quadrilateral embedding, in each of the following cases:

358 JOURNAL OF GRAPH THEORY

(i) m and r are both even; (ii) m and r are both odd, m 3 4 + r (mod 81, rn - r > 12.

We note that the graph G(m, m - 2) obtained by the composition met$od is Kmq minus a disconnected graph, whereas the current graph method, when it works, gives Km,m minus a Hamiltonian cycle. We also note that the genus of the constructed G(2n, 2n - 2h) is n2 - (h + 2)n + 1, and that, for all n 2 8m + 4, the genus of the constructed G(n, 8m t 4) is 2mn t 1. A suitable labelling of the vertices gives the following corollary.

Corollary 2.2. Let H(m, t) denote the bipartite graph on 16m + 2t + 8 vertices, regular of degree 8m + 4, with vertices a,, . . . , agm+t+4, b,, . . . , bgm+1+4, where ai is joined to bj except when j = i, i + 1,. . . , i t t - 1 (interpreted modulo 8m + t + 4). Then

y ( H ( m , t ) ) = 1 + 2m(8m + t + 4).

3. SOME SMALL GRAPHS

The assignment of currents to edges in Figure 2 is not of course unique. Different assignments can give different embeddings. Consider for example the current graph for n = 3 given in Figure 3(a). The resulting embedding of

on S,, has rotation scheme with initial row

0. 1 7 11 5 17 3 9 23 19 21 15 13,

and the quadrilateral regions of the embedding are

(a) ( 6 )

FIGURE 3. Currents in Z24

QUADRILATERAL EMBEDDINGS OF BIPARTITE GRAPHS 359

and all those derived from these three by adding i to each vertex (i = 1 , . . . , 23). Note that the differences between opposite vertices are 10,6; 4,12; 2,6. In all the quadrilaterals join odd opposite vertices differing by 2, 4, 6, 10, or 12, except that vertices which could in this way be joined more than once are joined only once. In the resulting graph, even vertices are joined to all odd vertices, and odd vertices are joined to all vertices except those differing by multiple of 8. Thus we have a embedding of K12,3,3,3,3 on S25. But this graph contains K12,12 as a subgraph, so that its genus cannot be less than 25, the genus of K12,12. We have therefore shown that

Next consider Figure 3(b) with currents again in Z24. The rotation scheme has first row

0. 1 5 15 1 1 19 1 7 21 23 13 7 3 9,

and the quadrilateral regions in the resulting embedding of K, , , on S25 are obtained from the following three “master” regions.

5 22 15 16 1 1 18

0 1 0 5 0 15

Note that the differences between opposite vertices are 2,4; 8,lO; 4,6. By joining all odd opposite vertices differing by 2,4, 6, 8, or 10, any two being joined only once as above, we see immediately that

(K12,2,2,2,2,2,2) = 25

and hence that

both graphs containing K,,,,, as a subgraph.

the familiar result that Alternatively, if we join a f f opposite vertices differing by 2,6, or 10, we get

Y ( K 6 , 6 , 6 , 6 ) = 25.

Jungerman [8] has shown that Y ( K ~ , ~ , ~ , ~ ) = ( n - 1)2 for all n # 3, the case n 2 (mod 4) being dealt with by trivalent current graphs.

360 JOURNAL OF GRAPH THEORY

FIGURE 4. Currents in ZZ5.

FIGURE 5. Currents in Z48.

4. DECOMPOSITIONS OF COMPLETE GRAPHS

The current graphs of Figure 4, with currents in Z2, , give in the standard way quadrilateral embeddings of two edge-disjoint graphs whose union is K25. By using r copies of G, we can easily generalize this to obtain a decomposition of K12r+I into r graphs each regular of degree 12 and each with a quadrilateral embedding on S12r+2. Using the notation of Cook [7] we thus have

N"(12r + 2 , . . ., 12r + 2) = 12r + 2,

where N*(yl , . . . , yr) denotes the smallest value of IZ for which K,, cannot be split up into r triangle-free subgraphs which are embeddable on orientable surfaces of genus yI , . . . , y,., respectively. This number is the triangle-free analog of the multiembedding number N(y, , . . . , yr) which Cook studies, following the study of N(yl, y2) in [3], [4], and [ 5 ] . In the light of the result N(38, 38) = 38 of [4], it is amusing to note thatN"(38, 38, 38) = 38. More generally, using r copies of Gb+], we find thatK4(2s+l )r+l can be split up into Y

graphs each with a quadrilateral embedding. Now consider decompositions of complete bipartite graphs. The current

graphs of figure 5 , with currents in Z48, give in the usual way quadrilateral embeddings in S4, of two bipartite graphs, regular of degree 12, whose union is K24,24. Thus t49(&4,24) = 2, where t,(G), the thickness of G on S,, is the smallest number of graphs embeddable on S, whose union is G. We note in passing that the two subgraphs of K24,24 given here are in fact isomorphic.

Using r copies of G3 we similarly obtain a decomposition o ~ K , ~ , , , ~ , into Y

bipartite graphs, regular of degree 12, each of which is quadrilaterally embeddable in S24r+I. Finally, using G2,+l instead of G3, we obtain a

QUADRILATERAL EMBEDDINGS OF BIPARTITE GRAPHS 361

decomposition of &,(23+1),4r(&+l) into r graphs each regular of degree 4(2s + 1) and each quadrilaterally embeddable on &~(2~+1)+1.

Theorem 5.1. ~8rs(&+1)+1(~4r(2s+I),4,(2s+1)) = r- In particular,K,(,+,,+ splits into two graphs of genus 16s(2s + 1) +

1.

ACKNOWLEDGMENT

The author wishes to thank the referee for his helpful comments.

References

[ 11 Y. Alavi, M. Jungerman, S . Stahl, and A. T. White, On imbeddings of hypergraphs. Graph Theory Newsl. 5 (1976) 108.

[2] S . R Alpert and J. L. Gross, Components of branched coverings of current graphs. J. Combinatorial Theory Ser. B 20 (1976) 283-303.

[3] I. Anderson and R. J. Cook, Bi-embeddings of graphs. Glasgow Math.

[4] I. Anderson and A. T. White, Current graphs and bi-embeddings. J. Graph Theory 2 (1978) 231-239.

[5] I. Anderson, Infinite families of bi-embedding numbers. J. Graph Theory 3 (1979) 263-268.

[6] A. Bouchet, Orientable and nonorientable genus of the complete bipartite graph. J. Combinatorial Theory Ser. B 24 (1978) 34-43.

[7] R J. Cook, unpublished. [8] M. Jungerman, The genus"of the symmetric quadripartite graph. J.

[9] T. Pisanski, Genus of Cartesian products of regular bipartite graphs. J.

[ 101 G. Ringel, Das Geschlecht des vollsthdigen paaren Graphen. Abh.

[ 111 G. Ringel, Map Color Theorem. Springer, Berlin (1974) [ 121 A. T. White, On the genus of the composition of two graphs. PuciJic J.

Math. 41 (1972) 275-279. 1131 A. T. White, Graphs, Groups and Surfaces. North-Holland,

Amsterdam (1973).

J . 15 (1974) 162-165.

Combinatorial Theory Ser. B 19 (1975) 181-187.

Graph Theory 4 (1 980) 3 1-42.

Math. Sem. Univ. Hamburg 28 (1965) 139-150.