On Voltage Graphs and Cages Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809 [email protected] 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 2g=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 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 possible2 orientations. If 1 e D(G), we denote the reverse arc by e− . If Γ is a finite group, a voltage2 assignment is a function f : D(G) Γ, such that if f(e) = g then 1 1 ! f(e− ) = g− . Given a voltage assignment, the lift of G in Γ via f, denoted Gf , 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 2 to the projection map πf : Gf G defined by πf (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 Gf be a lift of a graph G with voltage assignments in the group Γ. If v1; : : : ; vk are the vertices of a k-cycle in Gf , then πf (v1); : : : ; πf (vk) 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. 1When dealing with small abelian groups, we will typically think of the group additively. Hopefully this will cause no confusion. 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 Z2, the cyclic group of order two. We simplify the problem by considering a distinguished vertex v0 and the complete binary tree of radius two centered at v0, 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 g0 : : : g5 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 g0 : : : g5 are the elements of Z2 assigned to the six outer edges, we require that gi = 0 and gi + gi+1 + gi+2 = 0. Note that in this case 6 6 (Γ = Z2) the second condition is redundant. However, in general it may be needed. The assignment that meets these constraints is to let gi = 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 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 gi + gi+1 = 0. There are also three 6-cycles that contain opposite pairs of edges from6 the outer cycle. These edges are traversed in opposite orientations, so we require 5 that gi gi+3 = 0. A final 6-cycle is the outer cycle, so we have i=0 gi = 0. Summarizing,− 6 we have P 6 gi = 0 (1) 6 gi + gi+1 + gi+2 = 0 (2) 6 gi + gi+1 = 0 (3) 6 gi gi+3 = 0 (4) − 5 6 g = 0 (5) X i i=0 6 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 Z2 Z2. 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). 1 gig− = 1 (6) i+3 6 gigi+1gi+2gi+3 = 1 (7) 1 1 6 gigi+1g− g− = 1 (8) i+3 i+4 6 5 We will assign voltages from the symmetric group S3. The voltages for the girth 9 graph are assigned as given below. Note that we identify group elements with permutations on three symbols. (123) if i 0 mod 2 8 ≡ > (12) if i = 1 g = > i < (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 i2 = j2 = k2 = 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. 1 gigi+1g− = 1 (9) i+4 6 gigi+2gi+4 = 1 (10) 6 gigi+1g1+2gi+3gi+4 = 1 (11) 6 4 Girth 16 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.