All regular graphs of small odd order are vertex-magic

Transcription

1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 51 (2011), Pages All regular graphs of small odd order are vertex-magic J. S. Kimberley J. A. MacDougall School of Mathematical and Physical Sciences The University of Newcastle NSW 2308 Australia Abstract This paper deals with vertex-magic total labellings of graphs. Earlier work by many authors has shown many infinite families of graphs to admit such labelings. The fact that many of these graphs are regular led MacDougall to conjecture that all non-trivial regular graphs are vertexmagic. Previously Gray and MacDougall showed that all odd-order r- regular graphs (r 2) of order up to v = 19 are vertex-magic. In this paper, we report on computations that extend this range, to show that all odd-order r-regular graphs (r 2) of order up to v = 29 are vertexmagic. 1 Introduction A vertex-magic total labeling (VMTL) on a graph G(V,E) withv vertices and e edges is a one-to-one mapping λ from the vertices and edges onto the integers 1, 2,...,v+e so that the sum of the label on a vertex and the labels of its incident edges is constant, independent of the choice of vertex. This sum is called the magic constant and varies depending on which labels are assigned to the vertices and which to the edges. The second author has conjectured [4] Conjecture 1 With the exception of K 2 and 2K 3, all regular graphs have at least one VMTL. Based on McQuillan s work [6], Gray (in [2]) provided a major step toward the solution of this conjecture. He developed a procedure for beginning with a graph possessing a VMTL and adjoining an arbitrary 2-factor to produce a graph of the same order but larger size which also possesses a VMTL. Using that procedure, Gray and MacDougall showed [3] that for r 4everyr-regular graph of odd order v 17 possesses a VMTL. In this paper we report the results of computations that extend this result to orders v 29.

2 176 J.S. KIMBERLEY AND J.A. MACDOUGALL 2 Strong VMTLs Gray s method applies to those labelings which are strong, i.e. have the v largest labels assigned to the vertices (and thus the smallest possible magic constant). In [3] it is shown that the range of feasible values for the magic constant k of an r-regular graph is determined by vr 2 +2(v +1)(r +1) 4k vr 2 +2(v +1)(r +1)+2vr and for the labeling to be strong we must have k = 1 4 (vr2 +2(v +1)(r + 1)). This will only be possible if v is odd and r is even, or if v is a multiple of 4 and r is odd. Thus, for example, even cycles do not have strong VMTLs but odd cycles do (the standard caterpillar labeling is an example). Theorem 1 ([2, Theorem 2.1]) If G is a graph with a spanning subgraph H which possesses a strong VMTL and G E(H) is even-regular, then G also possesses a strong VMTL. The power of this theorem is illustrated by the case where the starting graph H is an odd cycle. Every odd cycle admits a strong VMTL and adjoining 2-factors repeatedly gives us the following result. Corollary 1 ([2]) Every Hamiltonian regular graph of odd order possesses a strong VMTL. 3 Strong VMTLs of 2-regular Graphs While it is true that asymptotically almost all regular graphs possess a spanning cycle (so that the corollary above applies), we must consider those that do not. Part of the investigation in [3] depended on the classical result of Petersen that every even-regular graph has a 2-regular spanning subgraph, in other words, is spanned by a disjoint union of cycles. Thus to prove that all even-regular graphs of odd order v are strongly vertex-magic, it is enough to show that all 2-regular graphs of order v have strong VMTLs. Table 1 below shows the number γ of 2-regular graphs for the relevant values of v; these numbers appear as the odd terms in sequence A in Sloane s On-Line Encyclopedia of Integer Sequences [9]. Table 1: The number of 2-regular graphs of order v and the strong magic constant k v k γ Some examples of strong VMTLs for 2-regular graphs are shown in Figures 1 and 2: those in the first figure are disjoint unions of distinct cycles, while the graph in

3 REGULAR GRAPHS OF SMALL ODD ORDER 177 the second figure has a repeated cycle. Not every odd-order 2-regular graph has a strong VMTL. It was noted in [3] that C 3 C 4 with order 7, 2C 3 C 5 with order 11 and 3C 3 C 4 with order 13 had no strong VMTL. This led the authors of [3] to wonder whether these were the initial examples in two infinite families of such graphs. Constructions by McQuillan [7] and our own computations now suggest that in fact these are the only counterexamples. Figure 1: Two Examples of Strong VMTLs Figure 2: Strong VMTL of 7C 3 C The tables presented in this paper contain examples of a strong VMTL for all the other 2-regular graphs of orders up to 29. These were discovered by extensive computer searches. In all the tables the graphs are listed in reverse lexicographic order on the set of cycle lengths. The examples in these tables thus constitute a proof of the following result: Theorem 2 Other than the three graphs, C 3 C 4, 2C 3 C 5 and 3C 3 C 4, every odd-order 2-regular graph of odd order less than 30 has a strong vertex magic total labelling. Proof. Tables 2, 3 and 9 to 22 list the edge label cycles of a strong VMTL for each graph other than the three exceptions.

4 178 J.S. KIMBERLEY AND J.A. MACDOUGALL Table 2: Strong VMTLs of the 2-regular graphs of odd order 13 v k 3 9 C 3 (1, 3, 2) 5 14 C 5 (1, 4, 2, 5, 3) 7 19 C 7 (1, 5, 2, 6, 3, 7, 4) C 3 C 4 no strong VMTL C 9 (1, 6, 2, 4, 7, 3, 9, 5, 8) 9 24 C 3 C 6 (1, 7, 2, 8, 6, 5)(3, 9, 4) C 4 C 5 (1, 7, 2, 5)(3, 8, 6, 4, 9) 3C 3 (1, 9, 5)(2, 7, 6)(3, 8, 4) C 11 (1, 7, 2, 5, 10, 6, 11, 3, 8, 4, 9) C 3 C 8 (1, 9, 8)(2, 6, 7, 4, 3, 11, 5, 10) C 4 C 7 (1, 10, 6, 7)(2, 5, 4, 11, 3, 9, 8) C 5 C 6 (1, 10, 7, 6, 2, 8)(3, 9, 5, 11, 4) 2C 3 C 5 no strong VMTL C 3 2C 4 (1, 10, 3, 11)(2, 6, 4, 5)(7, 9, 8) C 13 (1, 8, 12, 5, 13, 2, 11, 3, 9, 10, 6, 4, 7) C 3 C 10 (1, 8, 2, 12, 6, 13, 4, 7, 9, 11)(3, 10, 5) C 4 C 9 (1, 9, 8, 12)(2, 6, 13, 3, 11, 7, 4, 5, 10) C 5 C 8 (1, 10, 8, 12, 5, 9, 4, 11)(2, 7, 3, 13, 6) C 6 C 7 (1, 9, 8, 11, 3, 10)(2, 6, 12, 4, 5, 7, 13) 2C 3 C 7 (1, 9, 10)(2, 7, 13)(3, 5, 8, 4, 12, 6, 11) C 3 C 4 C 6 (1, 7, 10)(2, 8, 4, 5, 9, 11)(3, 12, 6, 13) C 3 2C 5 (1, 10, 4, 9, 11)(2, 7, 8)(3, 5, 12, 6, 13) 2C 4 C 5 (1, 11, 3, 10)(2, 6, 9, 8)(4, 12, 7, 13, 5) 3C 3 C 4 no strong VMTL As a result of the Theorem we can apply Gray s construction, adjoining whatever 2-factors we like to any of these initial 2-regular graphs. Thus we can prove our main result: Corollary 2 Other than the three graphs C 3 C 4, 2C 3 C 5 and 3C 3 C 4, every odd-order regular graph of order less than 30 possesses a strong VMTL. It was shown in [3] that other than these three exceptional graphs C 3 C 4,2C 3 C 5 and 3C 3 C 4, all even-regular graphs of orders 7, 11 and 13, are vertex-magic. The proof of this relied, in part, on the fact that every quartic graph is spanned by a 2-regular graph other than these three. If we now note that these three exceptions do admit VMTLs (just not strong ones) we obtain the main result of the paper: Corollary 3 Every odd-order regular graph of order less than 30 is vertex-magic.

5 REGULAR GRAPHS OF SMALL ODD ORDER 179 Table 3: Strong VMTLs of the 17 2-regular graphs of order 15 C 15 (1, 13, 9, 4, 6, 14, 7, 12, 11, 5, 10, 2, 15, 3, 8) C 3 C 12 (1, 9, 2, 12, 5, 4, 8, 11, 7, 15, 6, 14)(3, 10, 13) C 4 C 11 (1, 8, 14, 6, 10, 5, 7, 3, 15, 4, 13)(2, 9, 12, 11) C 5 C 10 (1, 14, 8, 3, 7, 11, 9, 5, 4, 12)(2, 10, 13, 6, 15) C 6 C 9 (1, 12, 11, 4, 10, 2, 15, 7, 9)(3, 6, 14, 5, 13, 8) 2C 3 C 9 (1, 12, 11, 4, 10, 8, 13, 3, 9)(2, 7, 15)(5, 14, 6) C 7 C 8 (1, 10, 11, 9, 14, 8, 2, 13)(3, 6, 7, 5, 12, 4, 15) C 3 C 4 C 8 (1, 10, 13)(2, 8, 14, 6, 3, 15, 4, 11)(5, 12, 9, 7) C 3 C 5 C 7 (1, 10, 11, 2, 13)(3, 6, 4, 12, 7, 5, 15)(8, 9, 14) 2C 4 C 7 (1, 10, 12, 9, 14, 2, 13)(3, 6, 11, 7)(4, 8, 5, 15) C 3 2C 6 (1, 13, 5, 11, 2, 9)(3, 6, 14, 7, 10, 12)(4, 8, 15) C 4 C 5 C 6 (1, 11, 9, 13)(2, 8, 15, 4, 14, 7)(3, 10, 6, 5, 12) 3C 3 C 6 (1, 11, 9)(2, 7, 14)(3, 12, 10)(4, 13, 5, 6, 8, 15) 3C 5 (1, 8, 15, 7, 14)(2, 10, 3, 11, 9)(4, 6, 13, 5, 12) 2C 3 C 4 C 5 (1, 8, 14)(2, 12, 11, 9)(3, 10, 7, 5, 13)(4, 6, 15) C 3 3C 4 (1, 10, 2, 13)(3, 7, 6, 15)(4, 5, 12)(8, 11, 9, 14) 5C 3 (1, 12, 11)(2, 8, 14)(3, 6, 15)(4, 13, 7)(5, 9, 10) 4 The Number of Strong VMTLs of 2-regular Graphs Our goal as reported in the last section was to discover a single strong VMTL for each 2-regular graph of odd order. This was enough to prove the existence of a VMTL for every regular graph of odd order v<31. However our computations have also provided more convincing evidence supporting Conjecture 1. It turns out that, apart from the small order graphs, there are very large numbers of VMTLs for almost all graphs under consideration. Tables 4 and 5 below contain the results of counting the number of strong VMTLs for the 2-regular graphs (these numbers appear as sequence A in Sloane s OEIS). We found the exact numbers for v 19, but the practical limit of complete counting turned out to be at v = 19, so in Tables 5 and 6 we present lower bounds for v =21andv = 23. The counts we found are strong evidence that every 2-regular graph is strongly vertex-magic, and thus that Conjecture 1 is true for even-order graphs. However we still have no good idea how to prove this.

6 180 J.S. KIMBERLEY AND J.A. MACDOUGALL Table 4: The number of strong VMTLs of 2-regular graphs of orders 3 to 17 A A Graph #svmtls C 3 1 C 5 1 C 7 9 C 3 C 4 0 C 9 31 C 3 C 6 8 C 4 C 5 4 3C 3 2 C C 3 C 8 81 C 4 C C 5 C C 3 C 5 0 C 3 2C 4 7 C C 3 C C 4 C C 5 C C 6 C C 3 C C 3 C 4 C C 3 2C C 4 C C 3 C 4 0 C C 3 C C 4 C C 5 C C 6 C C 3 C C 7 C C 3 C 4 C C 3 C 5 C C 4 C C 3 2C C 4 C 5 C C 3 C C C 3 C 4 C C 3 3C C 3 11 Graph #svmtls C C 3 C C 4 C C 5 C C 6 C C 3 C C 7 C C 3 C 4 C C 8 C C 3 C 5 C C 4 C C 3 C 6 C C 4 C 5 C C 3 C C 3 2C C 4 C 6 C C 5 C C 3 C 4 C C 5 2C C 3 C 5 C C 3 2C 4 C C 3 C 4 2C C 4 C C 3 C C 3 2C 4 174

7 REGULAR GRAPHS OF SMALL ODD ORDER 181 Table 5: The number of strong VMTLs of 2-regular graphs of orders 19 and 21 Graph #VMTLs C C 3 C C 4 C C 5 C C 6 C C 3 C C 7 C C 3 C 4 C C 8 C C 3 C 5 C C 4 C C 9 C C 3 C 6 C C 4 C 5 C C 3 C C 3 C 7 C C 4 C 6 C C 5 C C 3 C 4 C C 3 2C C 4 C 7 C C 5 C 6 C C 3 C 5 C C 3 2C 4 C C 5 2C C 6 C C 3 C 6 C C 3 C 4 C 5 C C 4 C C 3 C C 3 C 4 2C C 3 2C 5 C C 4 C 5 C C 3 C 4 C C 4 3C C 3 2C C 3 2C 4 C C 3 4C C 3 C 4 98 Graph #svmtls C C 3 C C 4 C C 5 C C 6 C C 3 C C 7 C C 3 C 4 C C 8 C C 3 C 5 C C 4 C C 9 C C 3 C 6 C C 4 C 5 C C 3 C C 10 C C 3 C 7 C C 4 C 6 C C 5 C C 3 C 4 C C 3 C 8 C C 4 C 7 C C 5 C 6 C C 3 C 5 C C 3 2C 4 C C 3 2C C 4 C 8 C C 5 C 7 C C 6 C C 3 C 6 C C 3 C 4 C 5 C C 4 C C 3 C C 5 2C C 6 C 7 C C 3 C 7 C C 3 C 4 C 6 C C 3 2C 5 C C 4 C 5 C C 3 C 4 C C C 3 C 4 2C C 3 C 5 C 6 C C 4 C 6 C C 4 2C 5 C C 3 C 5 C C 3 2C 4 C C 3 3C C 4 C 5 2C C 3 2C C 5 C C 3 C 4 C 5 C C 3 3C 4 C C 3 C C 3 3C C 3 2C 4 2C C 4 C C 3 C 4 C C 3 3C C 3 1

8 182 J.S. KIMBERLEY AND J.A. MACDOUGALL Table 6: The number of strong VMTLs of 2-regular graphs of order 23 Graph #VMTLs C C 3 C C 4 C C 5 C C 6 C C 3 C C 7 C C 3 C 4 C C 8 C C 3 C 5 C C 4 C C 9 C C 3 C 6 C C 4 C 5 C C 3 C C 10 C C 3 C 7 C C 4 C 6 C C 5 C C 3 C 4 C C 11 C C 3 C 8 C C 4 C 7 C C 5 C 6 C C 3 C 5 C C 3 2C 4 C C 3 C 9 C C 4 C 8 C C 5 C 7 C C 6 C C 3 C 6 C C 3 C 4 C 5 C C 4 C C 3 C C 3 2C C 4 C 9 C C 5 C 8 C C 6 C 7 C C 3 C 7 C C 3 C 4 C 6 C C 3 2C 5 C C 4 C 5 C C 3 C 4 C C 5 2C Graph #svmtls C 6 C 8 C C 3 C 8 C C 7 C C 3 C 4 C 7 C C 3 C 5 C 6 C C 4 C 6 C C 4 2C 5 C C 3 C 5 C C 3 2C 4 C C 7 2C C 3 C 4 2C C 3 C 5 C 7 C C 4 C 7 C C 3 2C 6 C C 4 C 5 C 6 C C 3 C 6 C C 5 C C 3 C 4 C 5 C C 3 3C 4 C C 3 C C 3 C 6 2C C 4 C 5 2C C 3 2C C 4 2C 6 C C 5 C 6 C C 3 C 4 C 6 C C 3 2C 5 C C 3 2C 4 C 5 C C 4 C C 3 C 4 C C 5 3C C 3 C 5 2C C 3 2C 4 2C C 3 C 4 2C 5 C C 4 C 5 C C 3 C 5 C C 3 2C 4 C C 3 4C C 4 3C C 3 C 4 2C C 3 3C 4 C C 3 C C 3 5C C 3 2C

9 REGULAR GRAPHS OF SMALL ODD ORDER 183 There is much stronger evidence for Conjecture 1, however. The data presented in this paper pertains to strong labelings, where the magic constant has the smallest feasible value. The very earliest studies of VMTLs of graphs led us to realize that values of the magic constant near the middle of the range of feasible values almost always allowed many more VMTLs than values near the extremes of the range, usually by several orders of magnitude. With a few small exceptions, the magic constant spectrum of a regular graph produces a classical bell-shaped distribution of counts (the symmetry is due to the duality of labelings for regular graphs [5]). We illustrate this with an example. The 3-cube Q 3 is a 3-regular graph of order 8 for which the feasible magic constant k lies in the range 36 k 48. We counted the number of VMTLs for each value of k and the results are shown in Table 7 below. Table 7: Distribution of number of VMTLs for Q 3 k N Σ Finally we wish to remind the reader how powerful Gray s Theorem is. Using it we have now proved that all regular graphs of odd order less than 30 have strong VMTLs. How many graphs are we talking about? Table 8 tabulates the number of r-regular graphs of odd order n as far as is known at present. For a fixed r the number grows exponentially with n, of course, and the numbers become enormous quickly. The italic entries in the table are new values we have calculated that were not known previously. These were calculated using Markus Meringer s algorithm [8] for fast generation of regular graphs. The numbers in the table are recorded as sequences A008483(n), A033301(n) and A165627(n) intheoeis.

10 184 J.S. KIMBERLEY AND J.A. MACDOUGALL Table 8: The number of r-regular graphs of order n. n r =2 r =4 r =6 r =8 r = ?? ???? ?? The calculations for this paper were done on a cluster of standard Windows PCs in student labs and written using the Magma package [1]. The VMTLs were found using branch-and-bound search methods which focussed on the size of the longest cycle in the graph. Several variations of the search procedure were used, depending on the cycle-lengths involved. Finally we remark that searching for a strong VMTL was easier than searching for an arbitrary one because there was only one suitable set of vertex labels, and hence also only one suitable set of edge labels.

11 REGULAR GRAPHS OF SMALL ODD ORDER 185 Table 9: Strong VMTLs of the 25 2-regular graphs of order 17 C 17 (1, 13, 2, 14, 10, 11, 9, 17, 8, 15, 3, 16, 6, 4, 7, 5, 12) C 3 C 14 (1, 15, 2, 9, 5, 14, 12, 3, 7, 6, 16, 4, 8, 17)(10, 11, 13) C 4 C 13 (1, 10, 8, 5, 17, 4, 11, 13, 6, 14, 12, 2, 15)(3, 9, 16, 7) C 5 C 12 (1, 15, 2, 12, 11, 14, 7, 17, 5, 13, 6, 9)(3, 10, 16, 4, 8) C 6 C 11 (1, 14, 12, 10, 15, 2, 11, 13, 5, 7, 9)(3, 16, 4, 17, 6, 8) 2C 3 C 11 (1, 14, 10)(2, 8, 6, 17, 4, 16, 3, 9, 7, 15, 11)(5, 13, 12) C 7 C 10 (1, 9, 8, 15, 11, 14, 6, 16, 5, 13)(2, 17, 7, 4, 12, 3, 10) C 3 C 4 C 10 (1, 13, 7, 17, 2, 10, 3, 15, 8, 9)(4, 11, 14, 12)(5, 6, 16) C 8 C 9 (1, 15, 11, 3, 7, 13, 2, 16)(4, 8, 17, 6, 5, 14, 10, 12, 9) C 3 C 5 C 9 (1, 15, 9, 14, 12, 7, 13, 2, 16)(3, 10, 11)(4, 8, 17, 5, 6) 2C 4 C 9 (1, 16, 2, 13, 6, 8, 17, 5, 15)(3, 9, 4, 7)(10, 14, 12, 11) C 3 C 6 C 8 (1, 9, 11, 4, 8, 13)(2, 14, 12, 5, 6, 16, 7, 17)(3, 10, 15) C 4 C 5 C 8 (1, 13, 4, 11, 9)(2, 14, 12, 6, 16, 5, 7, 17)(3, 10, 15, 8) 3C 3 C 8 (1, 13, 2, 17, 6, 5, 16, 9)(3, 14, 10)(4, 8, 12)(7, 15, 11) C 3 2C 7 (1, 13, 6, 17, 5, 10, 16)(2, 11, 14, 7, 3, 15, 9)(4, 8, 12) C 4 C 6 C 7 (1, 13, 2, 8, 3, 10, 16)(4, 15, 9, 12)(5, 7, 11, 14, 6, 17) 2C 5 C 7 (1, 13, 2, 10, 16)(3, 8, 11, 14, 7)(4, 12, 6, 17, 5, 15, 9) 2C 3 C 4 C 7 (1, 16, 8, 13)(2, 10, 15, 11, 9, 4, 14)(3, 12, 7)(5, 6, 17) C 5 2C 6 (1, 11, 8, 7, 13, 12)(2, 14, 3, 15, 9)(4, 10, 16, 5, 17, 6) 2C 3 C 5 C 6 (1, 12, 7, 14, 11)(2, 13, 3, 15, 9, 8)(4, 10, 16)(5, 6, 17) C 3 2C 4 C 6 (1, 11, 14, 12)(2, 16, 8, 9)(3, 13, 7)(4, 15, 6, 17, 5, 10) C 3 C 4 2C 5 (1, 15, 9, 2, 11)(3, 17, 8, 7)(4, 13, 10)(5, 14, 12, 6, 16) 3C 4 C 5 (1, 15, 2, 11)(3, 16, 8, 17)(4, 7, 14, 12, 6)(5, 9, 13, 10) 4C 3 C 5 (1, 15, 11)(2, 13, 12)(3, 14, 7)(4, 9, 10, 8, 16)(5, 6, 17) 3C 3 2C 4 (1, 17, 4, 12)(2, 13, 10)(3, 7, 15, 11)(5, 14, 6)(8, 9, 16)

12 186 J.S. KIMBERLEY AND J.A. MACDOUGALL Table 10: Strong VMTLs of the 39 2-regular graphs of order 19 C 19 (1, 12, 2, 17, 7, 18, 3, 9, 6, 5, 11, 15, 13, 14, 8, 10, 19, 4, 16) C 3 C 16 (1, 16, 10, 19, 9, 4, 17, 2, 13, 11, 3, 8, 14, 6, 12, 15)(5, 7, 18) C 4 C 15 (1, 11, 6, 17, 3, 19, 7, 18, 10, 4, 15, 12, 9, 2, 14)(5, 8, 16, 13) C 5 C 14 (1, 16, 7, 4, 11)(2, 12, 13, 9, 17, 3, 15, 6, 18, 10, 19, 8, 5, 14) C 6 C 13 (1, 10, 6, 18, 4, 19)(2, 15, 11, 17, 12, 13, 14, 7, 8, 5, 9, 3, 16) 2C 3 C 13 (1, 19, 4, 18, 6, 15, 11, 17, 12, 2, 16, 9, 10)(3, 14, 13)(5, 8, 7) C 7 C 12 (1, 13, 15, 14, 4, 17, 2, 9, 7, 10, 16, 11)(3, 12, 8, 5, 18, 6, 19) C 3 C 4 C 12 (1, 13, 3, 19, 6, 18, 5, 15, 14, 12, 16, 11)(2, 9, 4, 17)(7, 10, 8) C 8 C 11 (1, 10, 19, 2, 16, 11, 4, 8, 18, 7, 12)(3, 13, 15, 9, 5, 17, 6, 14) C 3 C 5 C 11 (1, 10, 19, 2, 16, 6, 14, 3, 13, 15, 12)(4, 11, 8)(5, 9, 17, 7, 18) 2C 4 C 11 (1, 10, 19, 2, 16, 4, 15, 11, 17, 5, 12)(3, 9, 14, 13)(6, 8, 7, 18) C 9 C 10 (1, 16, 13, 2, 10, 17, 3, 15, 6, 18)(4, 12, 11, 14, 8, 5, 9, 19, 7) C 3 C 6 C 10 (1, 16, 11, 2, 10, 8, 15, 14, 6, 18)(3, 12, 13)(4, 17, 5, 9, 19, 7) C 4 C 5 C 10 (1, 18, 6, 16)(2, 13, 5, 8, 15, 14, 11, 3, 17, 10)(4, 12, 9, 19, 7) 3C 3 C 10 (1, 18, 2, 10, 5, 9, 19, 7, 6, 16)(3, 8, 15)(4, 17, 12)(11, 14, 13) C 3 C 7 C 9 (1, 19, 6, 5, 11, 2, 13)(3, 15, 8, 4, 18, 9, 17, 12, 16)(7, 10, 14) C 4 C 6 C 9 (1, 13, 15, 3, 16, 10, 5, 6, 19)(2, 14, 7, 17, 12, 11)(4, 8, 9, 18) 2C 5 C 9 (1, 19, 6, 15, 13)(2, 10, 5, 14, 3, 8, 16, 7, 11)(4, 18, 9, 17, 12) 2C 3 C 4 C 9 (1, 13, 10, 16, 8, 7, 5, 6, 19)(2, 11, 17)(3, 15, 14)(4, 12, 9, 18) C 3 2C 8 (1, 12, 16, 13, 6, 15, 10, 17)(2, 18, 4, 11, 3, 8, 9, 14)(5, 19, 7) C 4 C 7 C 8 (1, 12, 2, 18, 4, 8, 17)(3, 13, 15, 14)(5, 19, 7, 16, 11, 10, 9, 6) C 5 C 6 C 8 (1, 17, 10, 5, 19, 7, 16, 12)(2, 9, 8, 4, 18)(3, 13, 6, 15, 14, 11) 2C 3 C 5 C 8 (1, 17, 2, 18, 4, 8, 3, 12)(5, 19, 7, 16, 9)(6, 11, 10)(13, 15, 14) C 3 2C 4 C 8 (1, 12, 11, 17)(2, 18, 4, 10, 6, 15, 14, 13)(3, 8, 9, 16)(5, 19, 7) C 5 2C 7 (1, 15, 9, 2, 13)(3, 17, 4, 8, 18, 5, 14)(6, 7, 11, 16, 12, 10, 19) 2C 6 C 7 (1, 15, 4, 9, 2, 13)(3, 14, 8, 18, 5, 7, 17)(6, 12, 16, 11, 10, 19) 2C 3 C 6 C 7 (1, 15, 9, 3, 17, 2, 13)(4, 14, 7)(5, 8, 18)(6, 11, 16, 12, 10, 19) C 3 C 4 C 5 C 7 (1, 15, 13)(2, 16, 11, 9)(3, 14, 10, 19, 6, 7, 12)(4, 8, 18, 5, 17) 3C 4 C 7 (1, 15, 2, 13)(3, 8, 18, 5, 17, 4, 9)(6, 14, 10, 19)(7, 12, 16, 11) 4C 3 C 7 (1, 15, 13)(2, 16, 10, 17, 4, 18, 11)(3, 9, 14)(5, 19, 6)(7, 12, 8) C 3 C 4 2C 6 (1, 17, 11, 14)(2, 10, 12, 8, 6, 15)(3, 16, 13)(4, 9, 18, 5, 19, 7) C 3 2C 5 C 6 (1, 17, 4, 12, 11, 14)(2, 15, 9, 18, 10)(3, 16, 13, 7, 19)(5, 8, 6) 2C 4 C 5 C 6 (1, 17, 8, 6, 14)(2, 15, 12, 10)(3, 16, 5, 18, 11, 13)(4, 9, 19, 7) 3C 3 C 4 C 6 (1, 17, 8, 14)(2, 10, 19, 7, 13, 15)(3, 16, 11)(4, 12, 9)(5, 18, 6) C 4 3C 5 (1, 11, 16, 6, 14)(2, 17, 9, 4, 12)(3, 15, 13, 8)(5, 18, 7, 10, 19) 3C 3 2C 5 (1, 11, 12, 2, 14)(3, 10, 19, 6, 16)(4, 7, 17)(5, 15, 13)(8, 9, 18) 2C 3 2C 4 C 5 (1, 11, 9, 4, 14)(2, 15, 12)(3, 13, 8, 16)(5, 6, 17)(7, 18, 10, 19) C 3 4C 4 (1, 10, 3, 18)(2, 12, 8, 16)(4, 11, 6, 19)(5, 17, 9, 7)(13, 15, 14) 5C 3 C 4 (1, 10, 15)(2, 11, 16)(3, 17, 12)(4, 13, 9, 19)(5, 14, 7)(6, 18, 8)

13 REGULAR GRAPHS OF SMALL ODD ORDER 187 Table 11: Strong VMTLs of the 60 2-regular graphs of order 21 C 21 (1, 18, 14, 13, 3, 15, 7, 17, 4, 11, 9, 5, 8, 21, 2, 10, 20, 6, 19, 12, 16) C 3 C 18 (1, 13, 17, 11, 8, 4, 12, 3, 21, 10, 19, 7, 6, 16, 2, 15, 5, 20)(9, 18, 14) C 4 C 17 (1, 16, 9, 15, 6, 14, 4, 11, 5, 7, 20, 3, 19, 13, 17, 2, 12)(8, 18, 10, 21) C 5 C 16 (1, 18, 5, 19, 8, 21, 7, 11, 3, 9, 17, 4, 13, 2, 20, 12)(6, 14, 16, 15, 10) C 6 C 15 (1, 11, 20, 12, 4, 14, 10, 15, 2, 13, 16, 6, 17, 3, 18)(5, 8, 19, 7, 21, 9) 2C 3 C 15 (1, 18, 3, 17, 15, 2, 13, 14, 12, 6, 16, 8, 5, 20, 11)(4, 10, 19)(7, 21, 9) C 7 C 14 (1, 18, 2, 14, 16, 10, 5, 12, 19, 13, 8, 6, 17, 11)(3, 21, 4, 9, 20, 7, 15) C 3 C 4 C 14 (1, 18, 2, 14, 17, 15, 3, 21, 4, 9, 13, 10, 19, 11)(5, 16, 12)(6, 20, 7, 8) C 8 C 13 (1, 12, 11, 15, 16, 4, 13, 5, 19, 10, 20, 7, 21)(2, 17, 8, 6, 9, 3, 18, 14) C 3 C 5 C 13 (1, 21, 7, 20, 10, 19, 13, 4, 8, 16, 9, 6, 12)(2, 17, 14)(3, 11, 15, 5, 18) 2C 4 C 13 (1, 12, 5, 19, 13, 8, 18, 11, 4, 10, 20, 7, 21)(2, 17, 6, 14)(3, 9, 16, 15) C 9 C 12 (1, 18, 14, 4, 17, 13, 9, 20, 11)(2, 12, 16, 7, 8, 19, 6, 10, 3, 21, 5, 15) C 3 C 6 C 12 (1, 11, 20, 9, 6, 19, 8, 14, 4, 12, 2, 18)(3, 21, 5, 16, 7, 10)(13, 17, 15) C 4 C 5 C 12 (1, 11, 20, 9, 6, 19, 8, 10, 3, 21, 5, 18)(2, 12, 16, 14)(4, 17, 15, 7, 13) 3C 3 C 12 (1, 18, 14, 2, 10, 5, 21, 3, 15, 7, 16, 12)(4, 13, 17)(6, 8, 19)(9, 20, 11) C 10 C 11 (1, 21, 8, 20, 3, 9, 5, 14, 13, 17, 15)(2, 19, 12, 6, 18, 7, 10, 16, 4, 11) C 3 C 7 C 11 (1, 21, 8, 20, 3, 17, 15)(2, 11, 14, 16, 10, 4, 13, 5, 7, 12, 19)(6, 9, 18) C 4 C 6 C 11 (1, 21, 8, 20, 3, 10, 7, 5, 9, 17, 15)(2, 13, 6, 18, 12, 19)(4, 14, 11, 16) 2C 5 C 11 (1, 21, 8, 20, 3, 11, 16, 10, 7, 5, 15)(2, 17, 13, 12, 19)(4, 9, 6, 18, 14) 2C 3 C 4 C 11 (1, 21, 8, 20, 3, 10, 16, 4, 13, 17, 15)(2, 12, 19)(5, 14, 11, 7)(6, 9, 18) C 3 C 8 C 10 (1, 21, 10, 8, 7, 19, 6, 15, 2, 12)(3, 13, 17, 11, 16, 4, 20, 9)(5, 18, 14) C 4 C 7 C 10 (1, 21, 10, 5, 16, 2, 12)(3, 13, 6, 14)(4, 20, 9, 18, 7, 19, 11, 17, 15, 8) C 5 C 6 C 10 (1, 12, 2, 10, 21)(3, 13, 5, 15, 8, 19, 7, 18, 14, 16)(4, 20, 9, 6, 11, 17) 2C 3 C 5 C 10 (1, 12, 2, 10, 21)(3, 18, 14, 16, 9, 20, 4, 15, 5, 13)(6, 11, 17)(7, 8, 19) C 3 2C 4 C 10 (1, 21, 10, 18, 14, 13, 3, 16, 2, 12)(4, 20, 9, 17)(5, 15, 8, 7)(6, 19, 11) C 3 2C 9 (1, 20, 5, 14, 2, 11, 6, 18, 13)(3, 9, 21, 7, 8, 10, 16, 4, 19)(12, 17, 15) C 4 C 8 C 9 (1, 13, 4, 15, 16, 10, 5, 20)(2, 11, 18, 14)(3, 19, 8, 12, 6, 17, 7, 21, 9) C 5 C 7 C 9 (1, 20, 5, 18, 6, 11, 2, 14, 13)(3, 19, 12, 17, 15)(4, 8, 7, 21, 9, 10, 16) 2C 6 C 9 (1, 13, 16, 10, 5, 20)(2, 11, 6, 17, 7, 21, 9, 18, 14)(3, 15, 4, 8, 12, 19) 2C 3 C 6 C 9 (1, 13, 2, 11, 5, 20)(3, 19, 8, 18, 14, 10, 7, 21, 9)(4, 15, 16)(6, 17, 12) C 3 C 4 C 5 C 9 (1, 13, 5, 20)(2, 11, 18, 6, 14)(3, 19, 12)(4, 8, 9, 21, 7, 16, 10, 17, 15) 3C 4 C 9 (1, 20, 5, 13)(2, 18, 14, 17, 7, 21, 9, 6, 11)(3, 19, 10, 16)(4, 8, 15, 12) 4C 3 C 9 (1, 20, 5, 19, 8, 4, 18, 2, 13)(3, 10, 16)(6, 12, 11)(7, 21, 9)(14, 17, 15) C 5 2C 8 (1, 20, 7, 16, 6, 14, 2, 12)(3, 9, 10, 5, 19, 11, 17, 15)(4, 21, 8, 18, 13) C 6 C 7 C 8 (1, 12, 2, 14, 16, 7, 20)(3, 9, 17, 15, 5, 19)(4, 11, 6, 13, 18, 10, 8, 21) 2C 3 C 7 C 8 (1, 20, 7, 11, 3, 19, 5, 12)(2, 18, 14)(4, 21, 8)(6, 17, 13, 15, 16, 10, 9) C 3 C 4 C 6 C 8 (1, 12, 7, 20)(2, 15, 16, 10, 18, 14)(3, 19, 5, 13, 17, 6, 9, 11)(4, 21, 8) C 3 2C 5 C 8 (1, 20, 7, 16, 10, 9, 6, 12)(2, 15, 13, 18, 14)(3, 11, 19, 5, 17)(4, 21, 8) 2C 4 C 5 C 8 (1, 12, 5, 19, 3, 11, 7, 20)(2, 10, 18, 14)(4, 21, 8, 15, 16)(6, 13, 17, 9) 3C 3 C 4 C 8 (1, 20, 7, 12)(2, 16, 14)(3, 11, 6, 9, 17, 15, 5, 19)(4, 21, 8)(10, 18, 13) 3C 7 (1, 11, 12, 19, 13, 7, 15)(2, 16, 14, 3, 18, 10, 17)(4, 20, 9, 6, 8, 5, 21) C 3 C 4 2C 7 (1, 11, 20, 4, 21, 5, 15)(2, 16, 12, 10, 19, 13, 17)(3, 14, 9, 18)(6, 8, 7) C 3 C 5 C 6 C 7 (1, 11, 6, 7, 13, 8, 15)(2, 17, 10, 18, 14, 16)(3, 19, 12)(4, 21, 5, 9, 20) 2C 4 C 6 C 7 (1, 11, 12, 15)(2, 16, 14, 17)(3, 19, 13, 7, 10, 18)(4, 20, 9, 6, 8, 5, 21) C 4 2C 5 C 7 (1, 11, 6, 7, 8, 12, 15)(2, 16, 14, 17)(3, 19, 13, 10, 18)(4, 21, 5, 9, 20) 3C 3 C 5 C 7 (1, 11, 20, 4, 21, 5, 15)(2, 12, 16)(3, 10, 19)(6, 17, 13)(7, 8, 9, 18, 14) 2C 3 2C 4 C 7 (1, 11, 17, 15)(2, 16, 3, 18)(4, 21, 5, 12, 19, 10, 20)(6, 8, 7)(9, 14, 13) C 3 3C 6 (1, 20, 8, 5, 7, 16)(2, 17, 3, 15, 12, 14)(4, 10, 21)(6, 18, 11, 19, 13, 9) C 4 C 5 2C 6 (1, 16, 7, 5, 8, 20)(2, 17, 9, 6, 18)(3, 11, 19, 13, 14, 15)(4, 12, 10, 21) 3C 3 2C 6 (1, 16, 3, 19, 8, 20)(2, 13, 5, 7, 6, 14)(4, 10, 21)(9, 17, 15)(11, 18, 12) 3C 5 C 6 (1, 20, 8, 6, 7, 16)(2, 17, 15, 5, 13)(3, 19, 11, 18, 9)(4, 12, 14, 10, 21) 2C 3 C 4 C 5 C 6 (1, 16, 6, 7, 8, 20)(2, 17, 12, 14)(3, 9, 18, 5, 15)(4, 10, 21)(11, 19, 13) C 3 3C 4 C 6 (1, 16, 11, 15, 8, 20)(2, 17, 12, 18)(3, 19, 13)(4, 14, 10, 21)(5, 9, 6, 7) 5C 3 C 6 (1, 20, 9, 3, 12, 16)(2, 18, 14)(4, 10, 21)(5, 8, 19)(6, 13, 17)(7, 11, 15) 2C 3 3C 5 (1, 16, 5, 15, 13)(2, 21, 8, 10, 17)(3, 9, 7, 6, 19)(4, 20, 11)(12, 18, 14) C 3 2C 4 2C 5 (1, 16, 14, 13)(2, 17, 15, 5, 21)(3, 19, 12)(4, 9, 20, 8)(6, 18, 7, 11, 10) 4C 4 C 5 (1, 13, 5, 16)(2, 21, 7, 17)(3, 19, 11, 20, 12)(4, 9, 18, 8)(6, 14, 15, 10) 4C 3 C 4 C 5 (1, 13, 16)(2, 21, 11, 17)(3, 9, 15)(4, 18, 8, 19, 12)(5, 10, 20)(6, 14, 7) 3C 3 3C 4 (1, 15, 8, 20)(2, 13, 17)(3, 11, 18, 14)(4, 9, 16)(5, 19, 7)(6, 12, 10, 21) 7C 3 (1, 18, 14)(2, 15, 16)(3, 13, 17)(4, 21, 8)(5, 19, 9)(6, 20, 7)(10, 11, 12)

14 188 J.S. KIMBERLEY AND J.A. MACDOUGALL Table 12: Strong VMTLs of the first 50 2-regular graphs of order 23 C 23 (1, 17, 2, 12, 19, 4, 20, 15, 5, 21, 7, 18, 9, 8, 13, 3, 10, 22, 11, 23, 6, 16, 14) C 3 C 20 (1, 16, 13, 7, 9, 23, 3, 15, 19, 4, 18, 6, 22, 5, 8, 17, 2, 12, 21, 14)(10, 11, 20) C 4 C 19 (1, 23, 6, 9, 5, 15, 17, 10, 20, 8, 18, 3, 22, 11, 2, 14, 4, 13, 21)(7, 12, 19, 16) C 5 C 18 (1, 13, 2, 16, 15, 9, 8, 19, 7, 23, 6, 22, 11, 21, 14, 20, 5, 18)(3, 17, 4, 12, 10) C 6 C 17 (1, 21, 9, 23, 5, 19, 16, 18, 11, 20, 13, 7, 14, 2, 15, 4, 22)(3, 12, 6, 8, 17, 10) 2C 3 C 17 (1, 21, 9, 23, 5, 14, 2, 15, 19, 16, 11, 20, 13, 8, 10, 4, 22)(3, 12, 17)(6, 7, 18) C 7 C 16 (1, 15, 3, 14, 10, 20, 11, 8, 7, 13, 19, 2, 12, 23, 6, 22)(4, 21, 5, 17, 16, 18, 9) C 3 C 4 C 16 (1, 15, 16, 18, 14, 4, 21, 5, 10, 3, 19, 2, 12, 23, 6, 22)(7, 17, 13, 20)(8, 9, 11) C 8 C 15 (1, 17, 16, 7, 6, 23, 8, 19)(2, 20, 15, 9, 21, 4, 22, 12, 5, 14, 18, 10, 11, 3, 13) C 3 C 5 C 15 (1, 17, 11, 16, 7, 10, 3, 13, 2, 20, 15, 18, 14, 5, 19)(4, 22, 12, 9, 21)(6, 23, 8) 2C 4 C 15 (1, 19, 8, 23, 6, 11, 5, 9, 21, 4, 22, 12, 7, 16, 17)(2, 20, 15, 13)(3, 10, 14, 18) C 9 C 14 (1, 12, 2, 20, 15, 17, 14, 9, 7, 8, 10, 23, 4, 16)(3, 22, 6, 13, 21, 5, 19, 11, 18) C 3 C 6 C 14 (1, 12, 2, 20, 15, 9, 7, 8, 11, 19, 10, 23, 4, 16)(3, 22, 6, 17, 14, 18)(5, 13, 21) C 4 C 5 C 14 (1, 12, 2, 20, 15, 19, 13, 17, 14, 9, 10, 23, 4, 16)(3, 22, 6, 18)(5, 11, 7, 8, 21) 3C 3 C 14 (1, 16, 4, 23, 10, 5, 19, 15, 3, 22, 6, 20, 2, 12)(7, 9, 14)(8, 11, 21)(13, 18, 17) C 10 C 13 (1, 20, 15, 2, 16, 9, 21, 13, 18, 14, 8, 7, 12)(3, 11, 5, 19, 10, 23, 4, 22, 6, 17) C 3 C 7 C 13 (1, 20, 15, 17, 7, 8, 6, 22, 4, 23, 10, 19, 12)(2, 16, 3, 14, 11, 5, 18)(9, 21, 13) C 4 C 6 C 13 (1, 20, 15, 2, 16, 3, 11, 13, 21, 9, 7, 8, 12)(4, 22, 6, 19, 10, 23)(5, 17, 14, 18) 2C 5 C 13 (1, 20, 15, 2, 16, 3, 17, 14, 8, 7, 18, 11, 12)(4, 22, 6, 10, 23)(5, 19, 13, 21, 9) 2C 3 C 4 C 13 (1, 20, 15, 16, 3, 19, 10, 23, 4, 22, 6, 11, 12)(2, 18, 14)(5, 13, 21, 9)(7, 17, 8) C 11 C 12 (1, 22, 7, 12, 23, 3, 13, 21, 6, 16, 2, 19)(4, 10, 20, 5, 8, 9, 15, 18, 14, 17, 11) C 3 C 8 C 12 (1, 19, 2, 16, 14, 17, 15, 10, 9, 5, 11, 22)(3, 12, 23)(4, 18, 6, 7, 20, 8, 21, 13) C 4 C 7 C 12 (1, 22, 12, 23, 3, 13, 15, 18, 6, 16, 2, 19)(4, 11, 21, 9, 20, 7, 10)(5, 8, 17, 14) C 5 C 6 C 12 (1, 22, 6, 16, 2, 19)(3, 13, 21, 9, 8, 5, 14, 17, 7, 20, 12, 23)(4, 10, 15, 18, 11) 2C 3 C 5 C 12 (1, 22, 6, 10, 15, 14, 20, 7, 17, 16, 2, 19)(3, 12, 23)(4, 13, 18)(5, 8, 11, 21, 9) C 3 2C 4 C 12 (1, 19, 2, 16, 9, 5, 14, 20, 11, 21, 8, 22)(3, 12, 23)(4, 18, 15, 13)(6, 7, 17, 10) C 3 C 9 C 11 (1, 17, 12, 23, 3, 20, 11, 4, 16)(2, 22, 10, 6, 8, 5, 14, 13, 21, 9, 19)(7, 18, 15) C 4 C 8 C 11 (1, 16, 4, 11, 5, 9, 18, 15, 14, 8, 17)(2, 19, 12, 23, 3, 20, 10, 22)(6, 13, 21, 7) C 5 C 7 C 11 (1, 16, 4, 11, 17)(2, 22, 10, 6, 21, 9, 5, 8, 14, 15, 19)(3, 20, 13, 18, 7, 12, 23) 2C 6 C 11 (1, 17, 14, 15, 4, 16)(2, 22, 10, 5, 11, 19)(3, 20, 8, 6, 7, 18, 9, 13, 21, 12, 23) 2C 3 C 6 C 11 (1, 16, 4, 18, 13, 15, 19, 2, 22, 10, 17)(3, 20, 9, 21, 12, 23)(5, 14, 11)(6, 8, 7) C 3 C 4 C 5 C 11 (1, 16, 4, 11, 17)(2, 19, 12, 23, 3, 20, 14, 13, 6, 10, 22)(5, 8, 21, 9)(7, 18, 15) 3C 4 C 11 (1, 17, 14, 16)(2, 22, 10, 19)(3, 20, 8, 6, 7, 18, 9, 13, 21, 12, 23)(4, 11, 5, 15) 4C 3 C 11 (1, 17, 16)(2, 22, 10, 20, 11, 5, 14, 15, 8, 6, 19)(3, 12, 23)(4, 18, 9)(7, 21, 13) C 3 2C 10 (1, 23, 10, 4, 12, 5, 15, 19, 11, 17)(2, 13, 22, 9, 16, 3, 18, 14, 8, 21)(6, 7, 20) C 4 C 9 C 10 (1, 23, 10, 20, 7, 19, 6, 15, 17)(2, 13, 22, 9, 11, 3, 14, 5, 8, 21)(4, 18, 16, 12) C 5 C 8 C 10 (1, 23, 10, 15, 17)(2, 13, 22, 9, 19, 11, 3, 18, 8, 21)(4, 12, 5, 14, 20, 7, 6, 16) C 6 C 7 C 10 (1, 23, 10, 18, 3, 14, 11, 19, 15, 17)(2, 13, 22, 9, 5, 8, 21)(4, 12, 7, 20, 6, 16) 2C 3 C 7 C 10 (1, 23, 10, 4, 12, 14, 8, 5, 15, 17)(2, 13, 22, 9, 20, 7, 21)(3, 18, 16)(6, 11, 19) C 3 C 4 C 6 C 10 (1, 23, 10, 20, 5, 17)(2, 13, 22, 9, 18, 14, 3, 11, 8, 21)(4, 12, 16)(6, 15, 19, 7) C 3 2C 5 C 10 (1, 23, 10, 3, 17)(2, 13, 22, 9, 19, 15, 11, 6, 8, 21)(4, 12, 20, 7, 18)(5, 16, 14) 2C 4 C 5 C 10 (1, 23, 10, 4, 17)(2, 13, 22, 9, 19, 15, 11, 5, 8, 21)(3, 14, 6, 16)(7, 18, 12, 20) 3C 3 C 4 C 10 (1, 23, 10, 17)(2, 15, 4, 16, 13, 22, 9, 5, 11, 21)(3, 12, 18)(6, 7, 19)(8, 20, 14) C 5 2C 9 (1, 19, 8, 15, 3, 23, 11, 2, 20)(4, 10, 5, 12, 21)(6, 13, 16, 14, 18, 17, 7, 9, 22) C 6 C 8 C 9 (1, 19, 16, 3, 23, 11, 2, 20)(4, 14, 15, 12, 21, 7, 8, 17, 13)(5, 18, 6, 10, 22, 9) 2C 3 C 8 C 9 (1, 20, 2, 11, 23, 3, 13, 19)(4, 15, 10, 7, 8, 21, 12, 16, 14)(5, 22, 9)(6, 18, 17) 2C 7 C 9 (1, 19, 13, 15, 3, 23, 11, 2, 20)(4, 10, 6, 18, 17, 12, 21)(5, 22, 9, 8, 7, 16, 14) C 3 C 4 C 7 C 9 (1, 19, 4, 15, 3, 23, 11, 2, 20)(5, 12, 21, 7, 9, 22, 10)(6, 18, 17, 8)(13, 16, 14) C 3 C 5 C 6 C 9 (1, 20, 2, 11, 23, 3, 15, 13, 19)(4, 12, 21, 8, 7, 10)(5, 22, 9, 16, 14)(6, 18, 17) 2C 4 C 6 C 9 (1, 19, 4, 15, 3, 23, 11, 2, 20)(5, 12, 21, 10)(6, 22, 7, 9, 16, 8)(13, 14, 18, 17)

15 REGULAR GRAPHS OF SMALL ODD ORDER 189 Table 13: Strong VMTLs of the remaining 38 2-regular graphs of order 23 C 4 2C 5 C 9 (1, 19, 16, 13, 3, 23, 11, 2, 20)(4, 14, 18, 10, 15)(5, 22, 9, 21, 12)(6, 8, 7, 17) 3C 3 C 5 C 9 (1, 19, 16, 15, 3, 23, 11, 2, 20)(4, 12, 21)(5, 9, 8, 7, 22)(6, 17, 13)(10, 14, 18) 2C 3 2C 4 C 9 (1, 19, 16, 14, 3, 23, 11, 2, 20)(4, 12, 21)(5, 13, 18)(6, 8, 7, 22)(9, 15, 17, 10) C 7 2C 8 (1, 14, 7, 21, 8, 18, 16, 15)(2, 20, 3, 11, 9, 4, 13, 17)(5, 19, 6, 12, 23, 10, 22) C 3 C 4 2C 8 (1, 14, 7, 17, 2, 20, 3, 15)(4, 16, 12, 23, 10, 22, 5, 9)(6, 11, 19)(8, 18, 13, 21) C 3 C 5 C 7 C 8 (1, 15, 3, 20, 2, 17, 14)(4, 9, 16)(5, 12, 23, 10, 22)(6, 8, 13, 21, 7, 19, 11, 18) 2C 4 C 7 C 8 (1, 14, 16, 18, 6, 7, 11, 15)(2, 20, 3, 17)(4, 13, 8, 21)(5, 9, 19, 12, 23, 10, 22) C 3 2C 6 C 8 (1, 14, 12, 23, 10, 22, 5, 15)(2, 20, 3, 21, 4, 17)(6, 7, 11, 19, 9, 8)(13, 18, 16) C 4 C 5 C 6 C 8 (1, 15, 16, 9, 4, 14)(2, 20, 3, 17)(5, 12, 23, 10, 22)(6, 8, 13, 21, 7, 19, 11, 18) 3C 3 C 6 C 8 (1, 14, 20, 2, 17, 13, 5, 15)(3, 11, 18)(4, 19, 12, 23, 10, 22)(6, 21, 7)(8, 9, 16) 3C 5 C 8 (1, 14, 11, 3, 15)(2, 20, 6, 7, 17)(4, 19, 9, 21, 8, 13, 18, 16)(5, 12, 23, 10, 22) 2C 3 C 4 C 5 C 8 (1, 15, 16, 14)(2, 17, 3, 11, 7, 18, 6, 20)(4, 19, 9)(5, 12, 23, 10, 22)(8, 13, 21) C 3 3C 4 C 8 (1, 15, 16, 14)(2, 18, 6, 17)(3, 19, 7, 11)(4, 13, 21)(5, 8, 20, 9, 12, 23, 10, 22) 5C 3 C 8 (1, 15, 14)(2, 21, 4, 16, 18, 3, 11, 17)(5, 13, 19)(6, 7, 20)(8, 22, 9)(10, 12, 23) C 3 C 6 2C 7 (1, 23, 8, 15, 17, 2, 14)(3, 19, 16, 4, 10, 18)(5, 21, 9, 20, 13, 12, 22)(6, 11, 7) C 4 C 5 2C 7 (1, 23, 8, 18, 17, 2, 14)(3, 19, 4, 10, 15)(5, 16, 12, 22)(6, 11, 21, 9, 20, 13, 7) 3C 3 2C 7 (1, 23, 8, 18, 17, 2, 14)(3, 19, 15)(4, 9, 21, 6, 22, 11, 10)(5, 12, 20)(7, 16, 13) C 4 2C 6 C 7 (1, 23, 8, 17, 2, 14)(3, 18, 15, 19)(4, 10, 13, 22, 5, 12, 16)(6, 7, 11, 21, 9, 20) 2C 5 C 6 C 7 (1, 23, 8, 17, 2, 14)(3, 18, 16, 13, 19)(4, 10, 7, 21, 9)(5, 22, 11, 12, 6, 20, 15) 2C 3 C 4 C 6 C 7 (1, 23, 8, 17, 2, 14)(3, 19, 15, 13, 22, 5, 18)(4, 10, 16)(6, 11, 7)(9, 20, 12, 21) 2C 3 2C 5 C 7 (1, 23, 8, 6, 17, 2, 14)(3, 19, 10)(4, 22, 12, 5, 16)(7, 11, 21, 9, 18)(13, 20, 15) C 3 2C 4 C 5 C 7 (1, 23, 8, 6, 17, 2, 14)(3, 19, 10)(4, 13, 5, 16)(7, 18, 9, 21)(11, 22, 12, 20, 15) 4C 4 C 7 (1, 23, 8, 5, 17, 2, 14)(3, 15, 11, 18)(4, 16, 7, 10)(6, 21, 9, 19)(12, 20, 13, 22) 4C 3 C 4 C 7 (1, 23, 8, 10, 17, 2, 14)(3, 18, 15, 11)(4, 19, 16)(5, 12, 20)(6, 22, 7)(9, 13, 21) C 5 3C 6 (1, 18, 12, 23, 4, 22)(2, 16, 6, 8, 13, 11)(3, 14, 19, 15, 17)(5, 10, 21, 7, 9, 20) 2C 3 C 5 2C 6 (1, 18, 12, 23, 4, 22)(2, 11, 3, 13, 8, 16)(5, 15, 17)(6, 9, 20, 14, 19)(7, 10, 21) C 3 2C 4 2C 6 (1, 18, 12, 23, 4, 22)(2, 16, 13, 20, 5, 11)(3, 19, 15, 17)(6, 8, 9)(7, 21, 10, 14) C 3 C 4 2C 5 C 6 (1, 18, 12, 23, 4, 22)(2, 19, 6, 11)(3, 17, 15)(5, 9, 7, 21, 10)(8, 16, 13, 20, 14) 3C 4 C 5 C 6 (1, 18, 12, 23, 4, 22)(2, 15, 3, 11)(5, 17, 16, 8, 20)(6, 14, 7, 9)(10, 19, 13, 21) 4C 3 C 5 C 6 (1, 22, 6, 16, 18)(2, 13, 11)(3, 17, 15)(4, 12, 23)(5, 9, 8, 21, 10, 20)(7, 14, 19) 3C 3 2C 4 C 6 (1, 22, 7, 18)(2, 19, 3, 11)(4, 12, 23)(5, 13, 21)(6, 14, 10, 20, 8, 9)(15, 17, 16) C 3 4C 5 (1, 12, 4, 13, 22)(2, 17, 14, 6, 16)(3, 11, 21, 8, 18)(5, 10, 23, 7, 20)(9, 15, 19) 2C 4 3C 5 (1, 12, 4, 13, 22)(2, 17, 5, 16)(3, 11, 20, 8, 21)(6, 14, 18, 9)(7, 23, 10, 15, 19) 3C 3 C 4 2C 5 (1, 12, 13, 22)(2, 17, 14)(3, 11, 21, 6, 15)(4, 18, 16)(5, 10, 23, 7, 19)(8, 9, 20) 2C 3 3C 4 C 5 (1, 12, 19, 13, 22)(2, 15, 14)(3, 16, 18)(4, 20, 6, 21)(5, 9, 11, 17)(7, 23, 10, 8) 6C 3 C 5 (1, 22, 12)(2, 17, 14)(3, 11, 21, 4, 18)(5, 15, 13)(6, 9, 20)(7, 23, 10)(8, 19, 16) C 3 5C 4 (1, 22, 6, 12)(2, 19, 13, 14)(3, 23, 8, 11)(4, 21, 9, 20)(5, 10, 7, 15)(16, 18, 17) 5C 3 2C 4 (1, 12, 22)(2, 19, 14)(3, 11, 21)(4, 13, 5, 15)(6, 20, 9, 16)(7, 23, 8)(10, 18, 17)

16 190 J.S. KIMBERLEY AND J.A. MACDOUGALL Table 14: Strong VMTLs of the first 65 2-regular graphs of order 25 C25 (1, 24, 3, 17, 18, 8, 14, 2, 16, 21, 12, 22, 7, 10, 4, 19, 13, 6, 9, 15, 23, 5, 25, 11, 20) C3 C22 (1, 21, 10, 25, 2, 16, 22, 11, 18, 3, 20, 6, 8, 17, 13, 4, 24, 12, 7, 9, 23, 14)(5, 15, 19) C4 C21 (1, 22, 15, 6, 10, 7, 11, 25, 2, 23, 3, 12, 19, 5, 14, 24, 8, 20, 9, 21, 13)(4, 18, 17, 16) C5 C20 (1, 23, 2, 19, 15)(3, 20, 18, 8, 6, 11, 25, 5, 10, 22, 9, 13, 7, 12, 17, 16, 21, 14, 4, 24) C6 C19 (1, 19, 9, 10, 22, 2, 25, 13, 4, 14, 7, 23, 8, 21, 5, 11, 3, 12, 24)(6, 16, 18, 15, 20, 17) 2C3 C19 (1, 24, 12, 3, 11, 7, 23, 8, 14, 20, 9, 10, 6, 22, 2, 25, 13, 4, 19)(5, 21, 16)(15, 18, 17) C7 C18 (1, 15, 2, 22, 3, 17, 18, 16, 6, 13, 23, 5, 9, 21, 10, 19, 7, 20)(4, 14, 24, 8, 25, 12, 11) C3 C4 C18 (1, 15, 2, 22, 3, 17, 18, 4, 14, 24, 8, 25, 12, 7, 19, 11, 16, 20)(5, 9, 6, 23)(10, 21, 13) C8 C17 (1, 13, 24, 6, 23, 8, 12, 4, 18, 3, 14, 20, 15, 21, 17, 2, 25)(5, 19, 9, 16, 7, 11, 22, 10) C3 C5 C17 (1, 25, 2, 17, 19, 15, 9, 11, 21, 14, 3, 18, 4, 12, 6, 24, 13)(5, 20, 8, 23, 10)(7, 16, 22) 2C4 C17 (1, 25, 2, 17, 19, 4, 12, 23, 8, 10, 5, 15, 9, 16, 6, 24, 13)(3, 14, 20, 18)(7, 22, 11, 21) C9 C16 (1, 20, 14, 12, 8, 23, 13, 2, 16, 22, 15, 17, 6, 21, 4, 18)(3, 11, 24, 9, 7, 10, 19, 5, 25) C3 C6 C16 (1, 20, 7, 19, 4, 18)(2, 13, 23, 8, 21, 17, 15, 22, 12, 5, 25, 3, 11, 24, 9, 16)(6, 14, 10) C4 C5 C16 (1, 18, 4, 21, 17, 10, 6, 14, 9, 24, 11, 3, 25, 5, 12, 20)(2, 16, 8, 23, 13)(7, 22, 15, 19) 3C3 C16 (1, 20, 18)(2, 16, 6, 19, 4, 12, 8, 23, 9, 24, 11, 3, 25, 5, 21, 13)(7, 17, 10)(14, 22, 15) C10 C15 (1, 20, 16, 2, 21, 13, 9, 17, 11, 19)(3, 14, 5, 10, 22, 15, 23, 12, 4, 25, 8, 6, 18, 7, 24) C3 C7 C15 (1, 20, 14, 3, 24, 7, 18, 6, 22, 10, 5, 9, 13, 17, 19)(2, 21, 16)(4, 25, 8, 11, 15, 23, 12) C4 C6 C15 (1, 20, 17, 19)(2, 16, 22, 8, 25, 4, 12, 23, 9, 5, 10, 18, 6, 13, 21)(3, 24, 7, 15, 11, 14) 2C5 C15 (1, 20, 18, 8, 25, 4, 12, 23, 7, 24, 3, 14, 11, 17, 19)(2, 16, 6, 13, 21)(5, 9, 15, 22, 10) 2C3 C4 C15 (1, 20, 6, 19)(2, 16, 22, 15, 17, 18, 12, 4, 25, 8, 14, 3, 24, 7, 21)(5, 10, 9)(11, 13, 23) C11 C14 (1, 22, 7, 23, 3, 12, 25, 13, 5, 11, 9, 24, 4, 18)(2, 19, 6, 8, 16, 20, 14, 21, 10, 17, 15) C3 C8 C14 (1, 22, 10, 17, 5, 11, 7, 18)(2, 19, 15)(3, 12, 25, 13, 16, 20, 4, 24, 9, 21, 14, 6, 8, 23) C4 C7 C14 (1, 18, 4, 24, 9, 16, 15, 2, 19, 17, 7, 20, 10, 22)(3, 12, 25, 13, 5, 11, 23)(6, 14, 21, 8) C5 C6 C14 (1, 22, 5, 11, 7, 17, 15, 2, 19, 16, 9, 24, 4, 18)(3, 12, 25, 13, 23)(6, 14, 20, 10, 21, 8) 2C3 C5 C14 (1, 18, 7, 11, 5, 17, 15, 2, 19, 16, 4, 24, 9, 22)(3, 12, 25, 13, 23)(6, 21, 8)(10, 20, 14) C3 2C4 C14 (1, 22, 10, 20, 16, 9, 24, 4, 23, 3, 12, 25, 13, 18)(2, 19, 15)(5, 17, 7, 11)(6, 14, 21, 8) C12 C13 (1, 17, 15, 21, 5, 16, 7, 23, 6, 10, 12, 2, 18)(3, 22, 9, 19, 8, 25, 13, 24, 11, 4, 20, 14) C3 C9 C13 (1, 17, 5, 16, 10, 19, 12, 2, 18)(3, 22, 6, 21, 15, 8, 25, 13, 24, 11, 4, 20, 14)(7, 23, 9) C4 C8 C13 (1, 17, 6, 10, 16, 5, 23, 14, 3, 22, 12, 2, 18)(4, 20, 9, 21, 15, 7, 24, 11)(8, 25, 13, 19) C5 C7 C13 (1, 17, 5, 16, 7, 23, 9, 22, 3, 14, 12, 2, 18)(4, 20, 8, 25, 13, 24, 11)(6, 10, 19, 15, 21) 2C6 C13 (1, 17, 6, 10, 21, 9, 23, 14, 3, 22, 12, 2, 18)(4, 20, 16, 5, 24, 11)(7, 19, 8, 25, 13, 15) 2C3 C6 C13 (1, 17, 10, 6, 23, 7, 24, 11, 4, 20, 12, 2, 18)(3, 14, 22)(5, 16, 21)(8, 25, 13, 9, 19, 15) C3 C4 C5 C13 (1, 18, 2, 12, 17)(3, 22, 10, 6, 24, 11, 4, 20, 7, 15, 19, 9, 14)(5, 21, 16)(8, 25, 13, 23) 3C4 C13 (1, 17, 14, 3, 22, 8, 25, 13, 24, 11, 12, 2, 18)(4, 23, 9, 20)(5, 10, 6, 16)(7, 19, 15, 21) 4C3 C13 (1, 17, 6, 9, 23, 4, 20, 10, 21, 16, 12, 2, 18)(3, 14, 22)(5, 24, 11)(7, 15, 19)(8, 25, 13) C3 C10 C12 (1, 17, 11, 25, 9, 18, 4, 12, 2, 19)(3, 22, 15, 23, 8, 24, 5, 10, 7, 16, 14, 21)(6, 13, 20) C4 C9 C12 (1, 17, 13, 20, 7, 16, 10, 18, 4, 12, 2, 19)(3, 22, 15, 23, 8, 24, 5, 14, 21)(6, 11, 25, 9) C5 C8 C12 (1, 19, 2, 12, 16, 14, 9, 25, 11, 4, 18, 17)(3, 22, 15, 23, 8, 24, 5, 21)(6, 13, 20, 7, 10) C6 C7 C12 (1, 19, 2, 12, 16, 11, 25, 9, 22, 3, 21, 17)(4, 13, 10, 6, 20, 15, 18)(5, 24, 8, 7, 23, 14) 2C3 C7 C12 (1, 17, 6, 21, 9, 25, 11, 15, 16, 12, 2, 19)(3, 13, 22)(4, 20, 18)(5, 14, 23, 10, 7, 8, 24) C3 C4 C6 C12 (1, 17, 14, 12, 2, 19)(3, 21, 16, 22)(4, 18, 5, 24, 8, 11, 25, 9, 6, 10, 7, 23)(13, 20, 15) C3 2C5 C12 (1, 17, 10, 7, 9, 25, 11, 22, 3, 12, 2, 19)(4, 18, 13, 6, 20)(5, 24, 8, 15, 23)(14, 16, 21) 2C4 C5 C12 (1, 19, 2, 12, 4, 18, 15, 13, 22, 3, 21, 17)(5, 24, 8, 23, 14)(6, 11, 25, 9)(7, 16, 10, 20) 3C3 C4 C12 (1, 17, 5, 24, 8, 7, 9, 25, 11, 12, 2, 19)(3, 16, 14)(4, 23, 10, 21)(6, 20, 18)(13, 22, 15) C3 2C11 (1, 13, 24)(2, 22, 6, 12, 3, 16, 5, 25, 4, 18, 14)(7, 20, 15, 8, 9, 11, 23, 10, 21, 17, 19) C4 C10 C11 (1, 24, 10, 21, 6, 15, 23, 14, 2, 22, 13)(3, 12, 20, 8, 9, 11, 7, 19, 17, 16)(4, 18, 5, 25) C5 C9 C11 (1, 24, 10, 21, 7, 11, 9, 8, 13)(2, 22, 5, 25, 4, 19, 17, 18, 15, 23, 14)(3, 12, 20, 6, 16) C6 C8 C11 (1, 24, 10, 11, 21, 7, 20, 13)(2, 22, 9, 8, 18, 4, 25, 5, 15, 23, 14)(3, 12, 6, 17, 19, 16) 2C3 C8 C11 (1, 13, 24)(2, 22, 14)(3, 12, 6, 17, 15, 20, 18, 4, 25, 5, 16)(7, 21, 10, 23, 11, 9, 8, 19) 2C7 C11 (1, 24, 10, 21, 11, 7, 13)(2, 22, 5, 25, 4, 19, 17, 18, 15, 23, 14)(3, 16, 6, 20, 8, 9, 12) C3 C4 C7 C11 (1, 13, 24)(2, 22, 6, 14)(3, 16, 5, 25, 4, 18, 9, 8, 15, 20, 12)(7, 11, 23, 10, 21, 17, 19) C3 C5 C6 C11 (1, 13, 24)(2, 22, 11, 20, 14)(3, 16, 7, 10, 8, 12)(4, 25, 5, 21, 6, 15, 23, 9, 19, 17, 18) 2C4 C6 C11 (1, 24, 9, 11, 21, 13)(2, 14, 23, 8, 20, 15, 6, 12, 3, 16, 22)(4, 18, 5, 25)(7, 19, 17, 10) C4 2C5 C11 (1, 13, 21, 7, 24)(2, 22, 5, 25, 4, 19, 17, 18, 15, 23, 14)(3, 12, 20, 6, 16)(8, 9, 11, 10) 3C3 C5 C11 (1, 13, 24)(2, 22, 14)(3, 16, 5, 25, 4, 18, 8, 20, 15, 19, 12)(6, 21, 17)(7, 10, 23, 9, 11) 2C3 2C4 C11 (1, 13, 24)(2, 22, 6, 15, 20, 16, 3, 12, 21, 17, 14)(4, 18, 5, 25)(7, 10, 8, 19)(9, 11, 23) C5 2C10 (1, 16, 15, 7, 12, 6, 8, 19, 2, 14)(3, 22, 4, 20, 17)(5, 18, 11, 21, 9, 24, 10, 25, 13, 23) C6 C9 C10 (1, 14, 2, 19, 17, 3, 22, 15, 16)(4, 18, 12, 11, 7, 20)(5, 23, 9, 24, 10, 25, 13, 6, 8, 21) 2C3 C9 C10 (1, 14, 2, 19, 11, 17, 6, 18, 4, 16)(3, 22, 15)(5, 9, 24, 10, 25, 13, 23, 8, 21)(7, 12, 20) C7 C8 C10 (1, 14, 2, 19, 18, 5, 23, 4, 20, 16)(3, 22, 9, 24, 10, 25, 13, 17)(6, 12, 7, 15, 11, 21, 8) C3 C4 C8 C10 (1, 16, 6, 8, 18, 19, 2, 14)(3, 17, 7, 22)(4, 15, 21, 9, 24, 10, 25, 13, 5, 23)(11, 12, 20) C3 C5 C7 C10 (1, 14, 2, 19, 9, 24, 10, 25, 13, 16)(3, 22, 5, 18, 4, 20, 17)(6, 12, 7, 23, 8)(11, 21, 15) 2C4 C7 C10 (1, 14, 2, 19, 4, 18, 12, 15, 21, 16)(3, 22, 7, 17)(5, 23, 9, 24, 10, 25, 13)(6, 8, 11, 20) C3 2C6 C10 (1, 14, 2, 19, 8, 18, 4, 15, 21, 16)(3, 22, 6, 23, 7, 17)(5, 9, 24, 10, 25, 13)(11, 12, 20) C4 C5 C6 C10 (1, 16, 21, 7, 11, 18, 12, 19, 2, 14)(3, 22, 5, 17)(4, 20, 6, 8, 15)(9, 24, 10, 25, 13, 23)

17 REGULAR GRAPHS OF SMALL ODD ORDER 191 Table 15: Strong VMTLs of the remaining 65 2-regular graphs of order 25. 3C3 C6 C10 (1, 14, 2, 19, 4, 23, 9, 11, 20, 16)(3, 22, 15)(5, 24, 10, 25, 13, 17)(6, 8, 18)(7, 21, 12) 3C5 C10 (1, 16, 21, 15, 4, 18, 12, 19, 2, 14)(3, 23, 9, 11, 22)(5, 24, 10, 25, 13)(6, 17, 7, 20, 8) 2C3 C4 C5 C10 (1, 14, 2, 19, 17, 20, 4, 22, 3, 16)(5, 18, 12, 15)(6, 23, 8)(7, 21, 11)(9, 24, 10, 25, 13) C3 3C4 C10 (1, 14, 2, 19, 12, 6, 21, 8, 20, 16)(3, 11, 22)(4, 18, 5, 15)(7, 23, 9, 17)(10, 25, 13, 24) 5C3 C10 (1, 16, 14)(2, 24, 10, 25, 13, 3, 18, 6, 17, 20)(4, 15, 21)(5, 23, 9)(7, 22, 11)(8, 19, 12) C7 2C9 (1, 16, 3, 15, 5, 25, 11, 4, 24)(2, 12, 22, 9, 23, 14, 19, 7, 20)(6, 10, 13, 8, 21, 17, 18) C3 C4 2C9 (1, 16, 3, 15, 5, 25, 11, 4, 24)(2, 12, 22, 9, 23, 6, 10, 13, 20)(7, 14, 21, 17)(8, 19, 18) 2C8 C9 (1, 16, 3, 15, 5, 25, 4, 24)(2, 20, 7, 8, 18, 6, 10, 11, 12)(9, 22, 13, 21, 17, 19, 14, 23) C3 C5 C8 C9 (1, 16, 3, 15, 5, 25, 11, 4, 24)(2, 12, 22, 7, 19, 8, 13, 20)(6, 10, 21, 17, 18)(9, 14, 23) 2C4 C8 C9 (1, 16, 3, 15, 5, 25, 4, 24)(2, 20, 18, 19, 8, 13, 22, 14, 12)(6, 9, 7, 17)(10, 21, 11, 23) C3 C6 C7 C9 (1, 16, 3, 15, 5, 25, 11, 4, 24)(2, 12, 20)(6, 18, 19, 8, 23, 10)(7, 22, 13, 21, 17, 9, 14) C4 C5 C7 C9 (1, 16, 3, 15, 5, 25, 11, 4, 24)(2, 12, 22, 7, 20)(6, 10, 23, 8, 13, 19, 18)(9, 14, 21, 17) 3C3 C7 C9 (1, 16, 3, 15, 5, 25, 11, 4, 24)(2, 12, 20)(6, 23, 10)(7, 14, 9, 22, 13, 21, 17)(8, 18, 19) C4 2C6 C9 (1, 16, 3, 15, 5, 25, 11, 4, 24)(2, 20, 7, 9, 22, 12)(6, 23, 10, 14, 21, 17)(8, 18, 19, 13) 2C5 C6 C9 (1, 16, 3, 15, 5, 25, 11, 4, 24)(2, 12, 22, 10, 13, 20)(6, 18, 19, 8, 23)(7, 14, 21, 17, 9) 2C3 C4 C6 C9 (1, 16, 3, 15, 5, 25, 11, 4, 24)(2, 12, 23, 10, 14, 20)(6, 21, 17)(7, 9, 22)(8, 13, 19, 18) 2C3 2C5 C9 (1, 16, 3, 15, 23, 9, 22, 4, 24)(2, 12, 21, 14, 20)(5, 25, 11)(6, 17, 10, 19, 18)(7, 13, 8) C3 2C4 C5 C9 (1, 16, 3, 15, 12, 2, 20, 4, 24)(5, 10, 23, 11, 25)(6, 14, 21, 17)(7, 22, 9)(8, 13, 19, 18) 4C4 C9 (1, 16, 3, 15, 5, 25, 13, 8, 24)(2, 20, 4, 12)(6, 9, 18, 17)(7, 21, 10, 19)(11, 22, 14, 23) 4C3 C4 C9 (1, 16, 3, 24)(2, 12, 8, 13, 10, 25, 5, 11, 20)(4, 14, 22)(6, 23, 9)(7, 21, 17)(15, 19, 18) C3 C6 2C8 (1, 17, 19, 4, 12, 22, 3, 16)(2, 24, 13, 9, 5, 25)(6, 14, 10, 11, 21, 7, 8, 23)(15, 20, 18) C4 C5 2C8 (1, 16, 3, 22, 14, 20, 18, 17)(2, 24, 13, 7, 9, 19, 5, 25)(4, 11, 21, 12, 10)(6, 15, 8, 23) 3C3 2C8 (1, 16, 17)(2, 24, 13, 22, 3, 19, 5, 25)(4, 10, 6, 23, 8, 7, 21, 15)(9, 11, 12)(14, 20, 18) C3 2C7 C8 (1, 16, 3, 22, 14, 7, 17)(2, 25, 5, 11, 21, 13, 24)(4, 19, 9, 6, 23, 8, 12, 10)(15, 20, 18) C4 C6 C7 C8 (1, 16, 3, 22, 12, 20, 18, 17)(2, 24, 13, 9, 11, 5, 25)(4, 19, 14, 10)(6, 15, 21, 7, 8, 23) 2C5 C7 C8 (1, 16, 3, 19, 17)(2, 24, 13, 7, 9, 5, 25)(4, 11, 10, 14, 21)(6, 22, 12, 20, 18, 15, 8, 23) 2C3 C4 C7 C8 (1, 17, 3, 16)(2, 24, 13, 19, 15, 10, 5, 25)(4, 20, 18)(6, 8, 23)(7, 21, 14, 22, 11, 12, 9) C5 2C6 C8 (1, 16, 3, 22, 10, 4, 19, 17)(2, 24, 13, 11, 5, 25)(6, 23, 8, 12, 9)(7, 21, 14, 20, 18, 15) 2C3 C5 C6 C8 (1, 16, 20, 14, 21, 17)(2, 24, 13, 7, 9, 19, 5, 25)(3, 12, 11, 10, 22)(4, 15, 18)(6, 23, 8) C3 2C4 C6 C8 (1, 17, 3, 16)(2, 24, 13, 11, 22, 10, 5, 25)(4, 12, 9, 19)(6, 23, 8)(7, 15, 21, 14, 20, 18) C3 C4 2C5 C8 (1, 17, 3, 16)(2, 24, 13, 9, 14, 19, 5, 25)(4, 12, 22, 10, 11)(6, 8, 23)(7, 21, 15, 20, 18) 3C4 C5 C8 (1, 17, 19, 16)(2, 24, 13, 21, 7, 9, 5, 25)(3, 12, 11, 10, 22)(4, 20, 18, 15)(6, 14, 8, 23) 4C3 C5 C8 (1, 16, 17)(2, 24, 13, 7, 9, 19, 5, 25)(3, 20, 18)(4, 15, 21, 14, 11)(6, 8, 23)(10, 12, 22) 3C3 2C4 C8 (1, 16, 17)(2, 24, 13, 23, 8, 14, 5, 25)(3, 12, 22)(4, 10, 19)(6, 15, 20, 18)(7, 9, 11, 21) C4 3C7 (1, 17, 18, 6, 16, 4, 24)(2, 19, 14, 13)(3, 11, 25, 7, 9, 10, 20)(5, 12, 22, 15, 23, 8, 21) C5 C6 2C7 (1, 17, 18, 16, 4, 24)(2, 19, 14, 12, 5, 9, 13)(3, 21, 6, 10, 20)(7, 22, 15, 23, 8, 11, 25) 2C3 C5 2C7 (1, 17, 16, 4, 24)(2, 13, 22, 15, 23, 8, 19)(3, 14, 20)(5, 9, 21)(6, 18, 11, 25, 7, 12, 10) C3 2C4 2C7 (1, 17, 10, 6, 18, 4, 24)(2, 19, 16, 13)(3, 14, 20)(5, 9, 11, 25, 7, 12, 21)(8, 22, 15, 23) 3C6 C7 (1, 17, 3, 20, 4, 24)(2, 19, 14, 21, 6, 16, 13)(5, 11, 25, 7, 10, 9)(8, 18, 12, 22, 15, 23) 2C3 2C6 C7 (1, 24, 4, 12, 10, 17)(2, 19, 11, 25, 7, 22, 13)(3, 21, 16)(5, 18, 15, 23, 8, 9)(6, 14, 20) C3 C4 C5 C6 C7 (1, 17, 3, 20, 4, 24)(2, 13, 21, 14, 19)(5, 9, 18, 12)(6, 10, 16)(7, 22, 15, 23, 8, 11, 25) 3C4 C6 C7 (1, 17, 12, 10, 4, 24)(2, 19, 16, 3, 14, 20, 13)(5, 21, 6, 18)(7, 9, 11, 25)(8, 22, 15, 23) 4C3 C6 C7 (1, 24, 4, 11, 25, 7, 17)(2, 19, 16, 6, 8, 18)(3, 13, 20)(5, 12, 22)(9, 10, 21)(14, 15, 23) C3 3C5 C7 (1, 17, 10, 4, 24)(2, 19, 18, 5, 21, 3, 13)(6, 16, 14)(7, 12, 22, 11, 25)(8, 9, 20, 15, 23) 2C4 2C5 C7 (1, 17, 10, 4, 24)(2, 19, 3, 16, 13)(5, 12, 21, 14, 20, 6, 18)(7, 9, 11, 25)(8, 22, 15, 23) 3C3 C4 C5 C7 (1, 24, 4, 18, 17)(2, 19, 14, 13)(3, 21, 16)(5, 12, 22, 7, 25, 11, 9)(6, 10, 20)(8, 15, 23) 2C3 3C4 C7 (1, 17, 5, 14, 10, 4, 24)(2, 19, 18)(3, 12, 21, 13)(6, 20, 15, 23)(7, 16, 11, 25)(8, 22, 9) 6C3 C7 (1, 17, 19)(2, 13, 22)(3, 14, 20)(4, 24, 5, 25, 6, 8, 18)(7, 12, 9)(10, 15, 23)(11, 21, 16) C3 C4 3C6 (1, 16, 22)(2, 20, 4, 15, 11, 23)(3, 18, 17, 10, 5, 13)(6, 8, 21, 9, 19, 14)(7, 24, 12, 25) C3 2C5 2C6 (1, 22, 14, 19, 16)(2, 23, 5, 9, 20)(3, 18, 6, 21, 17, 13)(4, 15, 11)(7, 24, 10, 8, 12, 25) 2C4 C5 2C6 (1, 16, 18, 3, 13, 22)(2, 20, 8, 6, 23)(4, 14, 19, 11, 9, 15)(5, 21, 17, 10)(7, 24, 12, 25) 3C3 C4 2C6 (1, 16, 22)(2, 20, 6, 8, 10, 23)(3, 21, 13)(4, 11, 19, 9, 18, 17)(5, 15, 14)(7, 24, 12, 25) C4 3C5 C6 (1, 22, 8, 10, 16)(2, 20, 15, 4, 23)(3, 13, 21, 17, 11, 18)(5, 9, 6, 14, 19)(7, 24, 12, 25) 3C3 2C5 C6 (1, 16, 22)(2, 20, 15, 5, 23)(3, 21, 13)(4, 14, 19, 17, 10)(6, 24, 7, 25, 12, 9)(8, 11, 18) 2C3 2C4 C5 C6 (1, 16, 22)(2, 23, 5, 10, 20)(3, 18, 17, 9, 11, 13)(4, 15, 19, 14)(6, 21, 8)(7, 24, 12, 25) C3 4C4 C6 (1, 22, 8, 11, 18, 16)(2, 20, 15, 23)(3, 21, 5, 13)(4, 10, 17)(6, 9, 19, 14)(7, 24, 12, 25) 5C3 C4 C6 (1, 16, 22)(2, 23, 4, 11, 18, 17)(3, 19, 15)(5, 21, 9)(6, 14, 10)(7, 24, 12, 25)(8, 13, 20) 5C5 (1, 16, 6, 25, 13)(2, 22, 3, 23, 14)(4, 24, 8, 10, 11)(5, 18, 17, 19, 15)(7, 12, 21, 9, 20) 2C3 C4 3C5 (1, 16, 6, 25, 13)(2, 21, 12, 18, 17)(3, 22, 14, 23)(4, 24, 8, 10, 11)(5, 15, 19)(7, 9, 20) C3 3C4 2C5 (1, 16, 6, 25, 13)(2, 19, 17)(3, 21, 14, 23)(4, 24, 8, 22, 12)(5, 18, 15, 10)(7, 20, 9, 11) 5C3 2C5 (1, 16, 6, 25, 13)(2, 19, 14)(3, 22, 15)(4, 23, 11)(5, 18, 17, 7, 21)(8, 12, 24)(9, 10, 20) 5C4 C5 (1, 16, 6, 25, 13)(2, 21, 3, 18)(4, 11, 14, 12)(5, 22, 7, 23)(8, 10, 9, 24)(15, 19, 17, 20) 4C3 2C4 C5 (1, 16, 6, 25, 13)(2, 19, 17)(3, 12, 23)(4, 21, 8, 24)(5, 22, 15, 18)(7, 9, 11)(10, 20, 14) 3C3 4C4 (1, 20, 17)(2, 18, 12)(3, 16, 13, 14)(4, 21, 5, 19)(6, 22, 10)(7, 15, 23, 8)(9, 24, 11, 25) 7C3 C4 (1, 20, 5, 17)(2, 12, 21)(3, 14, 24)(4, 22, 15)(6, 10, 18)(7, 23, 8)(9, 11, 25)(13, 19, 16)