On Voltage Graphs and Cages

Transcription

1 On Voltage Graphs and Cages Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN April 4, 2003 Abstract Using voltage graphs, we construct the the smallest known trivalent graph of girth 16. The graph has 960 vertices and is a lift of the Petersen graph. By way of preparation, other lifts of the Petersen graph are discussed. AMS Subject Classifications: 05C25, 05C35 1 Introduction We are interested in constructing regular graphs with specified degree and girth (recall that the girth of a graph is the length of a shortest cycle). A (k, g)-graph is regular graph of degree k and girth g, and a (k, g)-cage is a (k, g)-graph of minimum possible order. Define f(k, g) to be the minimum order among trivalent graphs with girth g. It is easy to show that { 2 g/2+1 if g is even f(g) = 3 2 (g 1)/2 2 if g is odd This lower bound on the size of cages is called the Moore bound. It is known that among the trivalent cages, the bound is achieved only for girths 5, 6, 8, and 12. The problem of finding cages has been chronicled by Biggs [2] and others. Those interested in more detail on the up-to-the-minute status of the problem can consult Royle s web site [12]. 1

2 2 The Petersen graph, denoted by P, is the smallest 3-regular graph with girth 5. In this paper, we investigate graphs that can be constructed as lifts of the P, and discover a new graph that is the smallest known trivalent graph of girth 16. The graph has 960 vertices, improving the old bound of 992 given in [1]. Our method is best described using voltage graph terminology, first introduced by Gross [8], which we now review. 2 Lifts and Voltage Graphs If G is a finite graph, denote its vertex and edge sets by V (G) and E(G). Also let D(G) denote the set of arcs on G. Each edge e E(G) is represented exactly twice in D(G), i.e., with each of the two possible orientations. If e D(G), we denote the reverse arc by e 1. If Γ is a finite group, a voltage assignment is a function f : D(G) Γ, such that if f(e) = g then f(e 1 ) = g 1. Given a voltage assignment, the lift of G in Γ via f, denoted G f, is the graph whose vertex set is V (G) Γ in which (u, g) and (v, h) are adjacent if uv E(G) and gf(uv) = h. We will also have occasion to refer to the projection map π f : G f G defined by π f (v, g) = v. In turns out that cycles in a lift correspond to a certain type of walk in the base graph, so we review some terminology related to walks. A walk in a graph is an alternating sequence of vertices and edges, beginning and ending with a vertex, in which each element in the sequence is incident with the elements before and after it. A closed walk is a walk that begins and ends with the same vertex. A non-reversing walk is a walk in which no vertex is preceded and followed by the same edge. Observation. Let G f be a lift of a graph G with voltage assignments in the group Γ. If v 1,..., v k are the vertices of a k-cycle in G f, then π f (v 1 ),..., π f (v k ) are the vertices of a closed non-reversing walk in G such that the product 1 of the voltage assignments along the walk is the group identity. 3 Lifts of the Petersen Graph We wish to investigate lifts of P with large girth, and begin by looking at four examples of lifts that have greater girth than does the Petersen graph. 1 When dealing with small abelian groups, we will typically think of the group additively. Hopefully this will cause no confusion.

3 3 Figure 1: The Petersen Graph Suppose we want to construct a lift of the Petersen graph of girth 6. Since the Heawood graph, of order 14, is the (3,6)-cage, the smallest graph we can hope for is a lift of order 20. This would entail using voltage assignments from Z 2, the cyclic group of order two. We simplify the problem by considering a distinguished vertex v 0 and the complete binary tree of radius two centered at v 0, and by assigning the group identity element to each edge in this tree. It then remains to assign voltages to the edges of a 6-cycle. We refer to the edges of the distinguished 6-cycle as outer edges, and to the remaining edges as tree edges. Let g 0... g 5 be the group elements assigned to these six edges in clockwise order (refer to Figure 1). Now P contains twelve 5-cycles, six of these consist of four tree edges and one outer edge, and six consist of two tree edges and three consecutive outer edges. So if g 0... g 5 are the elements of Z 2 assigned to the six outer edges, we require that g i 0 and g i + g i+1 + g i+2 0. Note that in this case (Γ = Z 2 ) the second condition is redundant. However, in general it may be needed. The assignment that meets these constraints is to let g i = 1 for 0 i < 6. So we have a lift of P with girth 6. Noting that P has no 7-cycles, we next construct a girth 8 lift. Since the 8-cage has 30 vertices, is the unique trivalent graph of girth 8, and is not a lift of P, the smallest candidate will have 40 vertices. In addition

4 4 to the constraints required in the girth 6 case, we need to insure that the lift contains no 6-cycles. P contains ten 6-cycles. Six of these contain two consecutive edges from the outer cycle, so we require that g i + g i+1 0. There are also three 6-cycles that contain opposite pairs of edges from the outer cycle. These edges are traversed in opposite orientations, so we require that g i g i+3 0. A final 6-cycle is the outer cycle, so we have 5 i=0 g i 0. Summarizing, we have g i 0 (1) g i + g i+1 + g i+2 0 (2) g i + g i+1 0 (3) g i g i+3 0 (4) 5 g i 0 (5) i=0 Inequalities (1) and (2) insure that there are no 5-cycles, inequalities (3), (4) and (5) eliminate 6-cycles. We will construct a girth 8 lift using voltage assignments from Z 2 Z 2. By assigning voltages with a first coordinate of 1 to all six outer edges, we eliminate 5-cycles. For the second coordinate, we assign alternating values of 0 and 1. This prevents 6-cycles. So the assignments are as follows. { (1, 0) if i 0 mod 2 g i = (1, 1) if i 1 mod 2 This graph is the only lift of P of order 40 and girth 8. Continuing in this vein, we observe that for girth 9, we need additional constraints to insure that there are no 8-cycles. The Petersen graph contains 15 cycles of length 8, such cycles being in one-to-one correspondences with the edges of P. One can see this correspondence by choosing an edge, and deleting all edges incident with either of the two vertices of the chosen edge. The remaining graph has a unique 8-cycle which is easily found since it consists of all edges incident with vertices of degree two. These 8-cycles require us to add the three conditions below (we now think of our groups multiplicatively). g i gi (6) g i g i+1 g i+2 g i+3 1 (7) g i g i+1 gi+3 1 g 1 i+4 1 (8)

5 5 We will assign voltages from the symmetric group S 3. The voltages for the girth 9 graph are assigned as given below. Note that we identify group elements with permutations on three symbols. g i = (123) if i 0 mod 2 (12) if i = 1 (13) if i = 3 (23) if i = 5 The resulting graph is sometimes called Foster s Graph and for some time it was the smallest known trivalent graph of girth 9. We now know that the 18 cages of girth 9 have 58 vertices [3, 5]. Finding the smallest lift of P with girth 10 is the last case that can be handled by hand. For this case we use the quaternion group of order 8. Recall that this group has elements 1, 1, i, i, j, j, k, and k satisfying i 2 = j 2 = k 2 = 1, ij = k, jk = i, and ki = j. The added constraints for girth 10 result from eliminating 9-cycles. Note that if we delete a vertex from P, along with the incident edges, the remain subgraph has exactly two 9-cycles. A close examination of these cycles reveals that we need three more constraints. 4 Girth 16 g i g i+1 g 1 i+4 1 (9) g i g i+2 g i+4 1 (10) g i g i+1 g 1+2 g i+3 g i+4 1 (11) It is not immediately clear that it is possible to construct lifts of P with large girth. In the case where the lifting group is abelian, the girth of the lift is limited. Recall that a theta graph is a graph consisting of two vertices of degree three joined by three independent paths. Theorem. Let G be a graph which contains a subgraph isomorphic to a theta graph. If the theta-subgraph contains t edges, then the girth of a lift of G via an abelian group is at most 2t. Proof. A theta graph determines a closed non-reversing walk in which each edge is traversed twice, once in each direction. Hence for any voltage assignment from an abelian group, the product along the walk is 1.

6 6 Since the Petersen graph contains theta graphs with eight edges, any lift via an abelian group has girth at most 16. In the remainder of this section we describe an example which achieves 16. The number of inequalities that the voltage assignments must satisfy is much larger for girth 16 that for the small examples discussed above. The table gives the numbers of closed non-reversing walks for each of the relevant lengths. Length Number Total 2357 Table. Closed non-reversing walks in the Petersen graph. So if we were trying to construct a lift of girth 16 using a nonabelian group, we would have a total of 2357 inequalities to satisfy (for abelian groups this number can be reduced by roughly 2/3). Our construction employs the group Z 48 Z 2. Once again we select a vertex v 0 V (P ) and assign the identity to all edges on the tree of radius two centered at v 0. It again remains to label the edges of the hexagon. Finding a solution using a computer search takes less than one second with a 1 ghz processor. Finding all solutions takes about a minute. One such solution is depicted in Figure 2. All solutions produce the same graph. 5 Lifts of Other Graphs In [7], we constructed the smallest known (3, 17)-graph. The construction can also be viewed as a lift. The base graph is shown in Figure 3. For

7 7 ( 2, 1) ( 1, 0) ( 5, 0) (34, 1) (44, 1) (35, 0) Figure 2: A trivalent graph of girth 16.

8 8 (0,15) u (0,27) (0,35) v (5,16) (5,39) (2,11) (2,33) Figure 3: Base graph for the (3,17)-graph. the lifting group we use the semi-direct product of Z 43 by Z 7. Since 43 1 mod 7, a nonabelian semi-direct product exists and the group operation is defined by (u 1, v 1 )(u 2, v 2 ) = (u 1 + v 1 mod 7, v 1 r u 2 + v 2 mod 43) where r satisfies r 7 1 mod 43 [9, 11]. We can take r = 4, since 4 7 = mod 43. The voltage assignments that produce the required lift are shown in Figure 3. Note that the group identity is assigned to all edges incident with the labeled vertices u and v. Other well-known graphs can be described as lifts. One of the nicest constructions among the current cage record holders is that of the (3, 19)- graph described in [10]. We can also describe this graph as lift using voltage assignments in the semi-direct product of Z 47 by Z 23. In this case, the base multigraph is shown in Figure 4. All edges incident with the vertex u (in

9 9 (9,0) u (2,1) (13,3) Figure 4: Base graph for Whitehead s (3,19)-graph. the figure) are assigned the identify, the other assignments are given in the figure. Our earlier construction of a (3, 14)-graph [6] can be viewed in the context of voltage graphs. Again, the base graph is actually a multigraph (see Figure 5) and the lifting group is a nonabelian group of order 48. This group is probably most easily identified as group number 18 in the GAP [13] catalog of groups for order 48.

10 10 Figure 5: Base graph for the (3,14)-graph. References [1] N.L. Biggs, Cubic Graphs with Large Girth, Combinatorial Mathematics: Proceedings of the Third International Conference, Annals of the New York Academy of Sciences 555, 1989, [2] N. Biggs, Constructions for cubic graphs of large girth. Electron. J. Combin., 5 (1998) A1. [3] N.L. Biggs and M.J. Hoare, A trivalent graph with 58 vertices and girth 9, Discrete Math 30 (1980) [4] L. Branković, M. Miller, J. Plesník, J. Ryan, J. Siráň. A note on constructing large Cayley graphs of given degree and diameter by voltage assignments, Electron. J. Combin., 5 (1998) R9. [5] G. Brinkmann, B. D. McKay and C. Saager, The smallest cubic graphs of girth nine, Combinatorics Probability and Computing, 5 (1995) [6] G. Exoo, A Small Trivalent Graph of Girth 14. Electron. J. Combin., 9 (2002) N3. [7] G. Exoo, Some Applications of pq-groups in Graph Theory, Discussiones Mathematicae Graph Theory, to appear.

11 11 [8] J. L. Gross, Voltage Graphs. Discrete Math, 9(1974) [9] M. Hall, Jr., The Theory of Groups, The Macmillan COmpany, new York, [10] M. Hoare. Triplets and Hexagons, Graphs and Combinatorics, 9 (1993) [11] J. J. Rotman, The Theory of Groups: An introduction, 3rd. ed., Allyn and Bacon, Inc., Boston, [12] G. Royle, (WWW document) gordon/data.html. [13] Martin Schönert et al, Groups, Algorithms and Programming, version 4 release 2. Department of Mathematics, University of Western Australia.