Resolvable Mendelsohn Triple Systems with Equal Sized Holes F. E. Bennett Department of Mathematics Mount Saint Vincent University Halifax, Nova Scoti

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1 Resolvable Mendelsohn Triple Systems with Equal Sized Holes F. E. Bennett Department of Mathematics Mount Saint Vincent University Halifax, Nova Scotia, Canada B3M 2J6 R. Wei Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln, NE, 68588, U.S.A. L. Zhu y Department of Mathematics Suzhou University Suzhou, P.R. China Abstract An HMTS of type f n 1 ; n 2 ; ; n h g is a directed graph DK n1 ;n 2 ;;nh which can be decomposed into 3-circuits. If the 3-circuits can be partitioned into parallel classes, then the HMTS is called an RHMTS. In this paper, it is shown that the RHMTSs of type m h exist when mh 0 (mod 3) and (m; h) 6= (1; 6), with the possible exception of h = 6 and m62 M 17, where M 17 = f m j m is divisible by a prime less than 17 g. The existence of Mendelsohn frames, which is closely related to RHMTS, is also considered in this paper. It is proved that a Mendelsohn frame of type t u exists if and only if u 4 and t(u?1) 0(mod 3) with 2 possible exceptions. Research supported by NSERC under grant A-5320 y Research supported by NSFC under grant

2 1 Introduction A complete multipartite directed graph DK n1 ;n 2 ;;nh is a graph with vertex set X = S 1ih X i; where X i (1 i h) are disjoint sets with jx i j = n i, such that any two vertices x and y from dierent sets X i and X j are joined by exactly two arcs (x, y) and (y, x). Suppose DK n1 ;n 2 ;;nh can be decomposed into 3 - circuits. The collection of all these 3 - circuits (called blocks) is denoted by B. We call (X; B) a holey Mendelsohn triple system, denoted by HMTS(v) where v = P h i=1 n i is called the order. Each set X i (1 i h) is called a hole and the multiset fn 1 ; n 2 ; ; n h g is called the type of the HMTS. We also use a i b j c k to denote the type, which means that in the multiset there are i occurrences of a, j occurrences of b, etc. The existence of HMTSs with uniform types is completely solved in [4]. If a set of blocks contains every vertex exactly once, then the set is called a parallel class. A decomposition is called resolvable, if all the blocks can be partitioned into parallel classes. A resolvable HMTS of order v is denoted by RHMTS(v). An RHMTS can be considered as a generalization of a resolvable Mendelsohn triple system (RMTS) which has been studied extensively (see [8]). In fact, an RMTS is equivalent to an RHMTS of type 1 v, the existence of which has been solved in [5]. On the other hand, an RHMTS can also be considered as a generalization of resolvable GDD (denoted RGDD). A GDD is a triple (X; G; B) which satises the following properties: 1. G is a partition of X into subsets called groups, 2. B is a family of subsets of X (called blocks) such that a group and a block 2

3 contain at most one common point, 3. every pair of points from distinct groups occurs in exactly blocks. The type of the GDD is the multiset f jgj : G 2 Gg. We shall also use an \exponential" notation to describe types. We use the notation (v, K, ) - GDD to denote the GDD when its set of block sizes is K and jxj = v. If M =f1g, then the GDD becomes a PBD. If K = fkg and with the type n k, then the GDD becomes a transversal design TD[k; n]. It is well known that the existence of a TD[k; n] is equivalent to the existence of k? 2 MOLS(n). If all blocks of a GDD can be partitioned into parallel classes, then the GDD is called resolvable GDD and denoted by RGDD. If we ignore the order of the points (vertices) in blocks, an RHMTS of order v becomes a (v, 3, 2) - RGDD. But the converse is not true, i.e., a (v, 3, 2)-GDD may not be orientable (see [8]). In this paper, we consider RHMTSs with equal-sized holes, i.e., n 1 = n 2 = = n h = m. In this case the type of the RHMTS will be denoted by m h. It is easy to see that a necessary condition for the existence of RHMTSs is that mh 0 (mod 3). One main result of this paper is to prove that the RHMTSs exist when mh 0 (mod 3) and (m; h) 6= (1; 6) with the possible exceptions of h = 6 and m 62 M 17 ; where M 17 = fm j m is divisible by a prime less than 17g. One conguration closely related to RHMTS is a Mendelsohn frame (see [3]). Let (X; B) be an HMTS. Suppose that there is a set P of partial parallel classes of B, which satises the following properties: 1. each P 2 P is a partition of X n X i for some i, where P B; 3

4 2. [ P 2P P = B. Then the design is called a Mendelsohnframe. The type of the Mendelsohn frame is the same as that of the HMTS. Mendelsohn frames have played an important role in constructing RHMTSs and their existence is of special interest. A necessary condition for the existence of a Mendelsohn frame of type t u is that u 4; t(u? 1) 0 (mod 3). In [3] it was proved that when u 4; t(u? 1) 0 (mod 3) and t 6 3 (mod 6), a Mendelsohn frame of type t u exists. In this paper, we also consider the existence of Mendelsohn frames. We shall prove that a Mendelsohn frame of type t u exists if and only if u 4 and t(u? 1) 0 (mod 3) with possible exceptions of type t 6, t 2 f3, 21g. 2 Constructions In this section, we display some recursive constructions of RHMTSs and Mendelsohn frames. These constructions are similar to the constructions used in [11]. Some known results of GDD and RGDD, which are related to these constructions, are also mentioned in this section. The rst construction about RHMTS involves using Mendelsohn frames and small RHMTSs to obtain larger RHMTS. We have the following recursive construction which is similar to [10, Theorem 2.2]. Construction 2.1. Suppose there is a Mendelsohn frame of type T, and let t 1 jt for every t 2 T. If for every t 2 T, there exists an RHMTS with type t t=t 1+1 1, then there is an RHMTS of type t u 1 where u = 1 + P t2t t=t 1. 4

5 To use Construction 2.1, we need some constructions of Mendelsohn frames. The following three lemmas are from [3]. Lemma 2.2. Let (X; G; A) be a GDD, and let w: X! Z + [f0g (This is called weighting). Suppose there is a Mendelsohn frame of type fw(x)jx 2 A g for each A 2 A. Then there is a Mendelsohn frame of type f P x2g w(x)jg 2 Gg. Lemma 2.3. Suppose there is a Mendelsohn frame of type T, and suppose there is an RTD(3, n). There is a Mendelsohn frame of type fntjt 2 T g. Lemma 2.4. Let t be a positive integer, u 4 and t(u-1) 0 (mod 3). If t 6 3 (mod 6), then there exists a Mendelsohn frame of type t u. The following construction of RHMTS or Mendelsohn frame is analogous to [11, Theorem 2.10], so the proof is ommited. Construction 2.5. If there exists an RHMTS (Mendelsohn frame) of type t u ; t = t 1 h and an RHMTS (Mendelsohn frame) of type t h 1, then there exists an RHMTS (Mendelsohn frame) of type t uh 1. Now we state some known results about the existence of RGDDs, GDDs and RMTSs which will be used in the next two sections. Theorem 2.6. [10, 1, 9] A (v, 3, 1)-RGDD of type m u exists if and only if m(u? 1) 0 (mod 2); mu 0 (mod 3) and m u 6= 2 3 ; 2 6 and

6 Theorem 2.7. [5] An RMTS(v) exists if and only if v 0 (mod 3), v 6= 6. Theorem 2.8. [7] A (v, 4, 1) - GDD of type g u exists if and only if g(u?1) 0 (mod 3), g 2 u(u? 1) 0 (mod 12) and (g; u) 62 f(2; 4); (6; 4)g. 3 Existence of RHMTSs From a (v, 3, 1)-RGDD we can easily obtain an RHMTS of the same type. In fact, if every block fa, b, cg of the RGDD is substituted by two 3-circuits (also called cyclically ordered blocks) (a, b, c) and (b, a, c), the RGDD becomes an RHMTS of the same type. Lemma 3.1. If m is even or h is odd and mh 0 (mod 3), then there exists an RHMTS of type m h. Proof When m h 6= 2 3 ; 2 6 or 6 3, the conclusion comes from Theorem 2.6. The three cases not covered by Theorem 2.6 are displayed as follows: Type 2 3 points: f a i ; b j ; c k j1 i; j; k 2 g holes: ffx 1 ; x 2 gjx = a; b; cg parallel classes: (a 1 ; b 2 ; c 1 ) (a 2 ; c 2 ; b 1 ); (a 1 ; b 1 ; c 2 ) (a 2 ; c 1 ; b 2 ); (a 1 ; c 1 ; b 1 ) (a 2 ; b 2 ; c 2 ); (a 1 ; c 2 ; b 2 ) (a 2 ; b 1 ; c 1 ): Type 2 6 6

7 points: Z 10 [ f1 1 ; 1 2 g holes: ff i, i+5gj0 i 4g [ f1 1 ; 1 2 g parallel classes: (0; 2; 3) (6; 9; 7) (1; 5; 1 1 ) (8; 4; 1 2 ) (mod 10): Type 6 3 points: Z 12 [ f1 i j1 i 6g holes: ffi, i+2, i+4, i+6, i+8, i+10gji = 1; 2g [ f1 j j1 j 6g parallel classes: (0; 3; 1 1 ) (9; 2; 1 2 ) (4; 11; 1 3 ) (5; 6; 1 4 ) (8; 7; 1 5 ) (1; 10; 1 6 ) (mod 12): From now on we consider the case when m is odd and h is even. We shall divide this case into two sub-cases: m 0 (mod 3) and m 6 0 (mod 3). First we treat the case m 0 (mod 3). So we consider the RHMTSs with hole size 3. Lemma 3.2. There exist RHMTSs of type 3 h for h 2 f4, 6, 8, 10, 14g. Proof For h = 6, the design comes from [5]. For other values of h, see Appendix A. Lemma 3.3. If there exists an RHMTS of type t u, then there exists an RHMTS of type(mt) u for any positive integers m, where m 6= 2, 6. Proof For any positive integer m, m 6= 2, 6 there is an RGDD of type m 3 by 7

8 Theorem 2.6. Let the groups of the RGDD be fig M, where 1 i 3 and M is an m-set. Denote the block set by A. For the given RHMTS of type t u, let the holes be fxg T, where x belongs to a u-set U and T is a t-set. Let the block set of the RHMTS be B. We may construct an RHMTS of type (mt) u as follows. The holes are fxg(t M); x 2 U: For each block B = ((x; a); (y; b); (z; c)) 2 B and block A = f(1; m 1 ); (2; m 2 ); (3; m 3 )g 2 A, dene B(A) = ((x; a; m 1 ); (y; b; m 2 ); (z; c; m 3 )). Denote B(A) = [ A2A B(A) and D = [ B2B B(A). It is not dicult to see that D is the block set of an HMTS of type (mt) u : Let P(A) and P(B) denote the parallel classes of the RGDD and RHMTS respectively. For each P 2 P(A) and Q 2 P(B), it is clear that R(P; Q) = fb(a)jb 2 Q; A 2 P g is a parallel class of the resultant HMTS. Since D = [ P 2P(A);Q2P(B) R(P; Q); the resultant HMTS is indeed resolvable. Lemma 3.4. If there exist RHMTSs of type 3 t and 3 m, where m 6=2 or 6, then there exists an RHMTS of type 3 tm. Proof From an RHMTS of type 3 t we obtain an RHMTS of type (3m) t by Lemma 3.3. So we can use the RHMTS of type 3 m to obtain an RHMTS of type 3 mt by Construction 2.5. Lemma 3.5. There exist RHMTSs of type 3 h for h 2 f12, 16, 18, 22, 28, 30g. Proof From a Mendelsohn frame of type 1 7 and RTD(3, 9) we obtain a Mendel- 8

9 sohn frame of type 9 7 by Lemma 2.3. Then we obtain an RHMTS of type 3 22 by letting t = 9 and t 1 = 3 in Construction 2.1. For the other values of h we make use of Lemma 3.4 by taking m and t as follows: h t m The existence of RMTSs of type 3 3 ; 3 5 ; 3 7 existence of RMTSs of type 3 4 ; 3 6 comes from Lemma 3.2. comes from Lemma 3.1, while the Lemma 3.6. There exist Mendelsohn frames of type (3t) 4 (3l) 1, where t 6= 2, 3, 6 and 10, 0 l t. Proof For positive integers not equal to 2, 3, 6 and 10, there exist three MOLS of order t. Deleting some points of one group we obtain a GDD of type t 4 l 1 with block size 4 or 5. Give every point weight 3. Using Mendelsohn frames of types 3 4 and 3 5 as input designs (which are from Lemma 2.4), we obtain the desired Mendelsohn frame from Lemma 2.2. From Lemma 3.6 we obtain the following recursive construction of RHMTSs with hole size 3. Lemma 3.7. If there exist RHMTSs of type 3 t+1 and 3 l+1, then an RHMTS of type 3 4t+l+1 exists, where t 6= 2, 3, 6, 10 and 0 l t. Proof Use the Mendelsohn frame described in Lemma 3.6 we obtain the desired RHMTS by Construction 2.1. Lemma 3.8. For any h, h 3, there exists an RHMTS of type 3 h. Proof When h is odd the conclusion follows from Lemma 3.1. For even h 9

10 2 f4; 6; ; 18; 22; 28; 30g, a Mendelsohn frame of type 3 h exists from Lemmas 3.2 and 3.5. For other even h, we apply Lemma 3.7 with (t; l) 2 f(4, 3), (5, 3), (5, 5), (7, 3), (7, 5), (8, 3), (8, 5), (2j+1, i) j i = 3, 5, 7, 9 and j 4g. Now we can settle the case of m 0 (mod 3). Lemma 3.9. For m 0 (mod 3) and h 3, There exist RHMTSs of type m h. Proof Let m = 3t. If t is even, an RHMTS of type m h exists by Lemma 3.1. If t is odd, we can construct the desired RHMTS from an RHMTS of type 3 h by Lemma 3.3. Next we consider the case m 6 0 (mod 3). In this case we must have h 0 (mod 3). Lemma For h 0 (mod 3) and h 6= 6, there exist RHMTSs of type m h. Proof From Theorem 2.7 we have an RMTS of order h, i.e. an RHMTS of type 1 h. So when m 6= 2,6, we obtain the desired RHMTS by Lemma 3.3. But when m = 2 or 6, the design is given by Lemma 3.1. For the case where h = 6 and m is odd, the situation seems more dicult. We only obtain the following lemma. Lemma For m 2 M 17 there exist RHMTSs of type m 6, where M 17 = fm j m is divisible by a prime less than 17g. Proof Let m be a prime less than 17. When m = 2 and 3, the RHMTSs of of type m 6 come from Lemma 3.1 and Lemma 3.2. For other values of m, 10

11 the RHMTSs are constructed in Appendix A. So the conclusion follows from Lemma 3.3. Lemma There does not exist any RHMTS of type 1 6. Proof The conclusion comes from Theorem 2.7 directly. Combining Lemmas 3.1, 3.9, 3.10, 3.11 and 3.12 we obtain our main theorem of this section which is stated below. Theorem An RHMTS of type m h exists if and only if mh 0 (mod 3) and (m; h) 6= (1,6) with possible exceptions of h = 6 and m 62 M 17, where M 17 = f m j m is divided by a prime less than 17 g. 4 Existence of Mendelsohn frames Now we turn attention to the existence of Mendelsohn frames. From Lemma 2.4 we have the following lemma. Lemma 4.1. If u 1(mod 3), then there exists a Mendelsohn frame of type 1 u. In view of Lemma 2.4 we need consider Mendelsohn frames of type t u with t 3 (mod 6), so we rst construct Mendelsohn frames of type 3 u. Lemma 4.2. Mendelsohn frames of type 3 u exist when u 1 (mod 3). Proof Since there exists an RTD(3,3) and a Mendelsohn frame of type 1 u, the 11

12 conclusion follows from Lemma 2.3. Lemma 4.3. Mendelsohn frames of type 3 u exist when u 0 or 1 (mod 4). Proof Since there exist (3u, 4, 1)-GDD of type 3 u by Theorem 2.8 and Mendelsohn frame of type 1 4 by Lemma 4.1, the conclusion follows from Lemma 2.2. Lemma 4.4. Mendelsohn frames of type 3 u exist when u 3 (mod 4) is a prime power. Proof Let be a primitive root of GF(u). We then form a Mendelsohn frame of type 3 u on the point set GF (u)z 3 by developing the base blocks f (( 2i,0), ( 2i (1 + ),1), ( 2i+1,0)), ((? 2i,2), (? 2i (1 + ),1), (? 2i+1,2)) j0 i u?3 2 g modulo (GF(u),3). Since the base blocks represent a partial parallel class, then all blocks developed form a Mendelsohn frame. Lemma 4.5. Mendelsohn frames of type 3 u exist for u 2 f14, 15, 18, 26, 30g. Proof From Appendix B we have Mendelsohn frames of type ; ; ; and from Lemmas 4.3 we have Mendelsohn frames of type 3 4 ; 3 5 ; 3 8, so we can easily obtain Mendelsohn frames of type 3 u for u 2 f14, 15, 18, 26g. From Theorem 2.8 and Lemma 4.1 we have a Mendelsohn frame of type We add 6 extra elements to this Mendelsohn frame as follows. For the rst three holes of the frame of type 21 4, we construct a Mendelsohn frame of type (which is from Appendix B) on the elements of every hole and the extra elements such that the hole of size 6 is the extra elements. For the fourth hole, we construct a Mendelsohn frame of type 3 9 (which is from Lemma 4.3) on the elements of that hole and the extra elements. It is readily to check that 12

13 all the blocks obtained form a Mendelsohn frame of type Now we need some result about PBDs. The following lemma is well known, for example see [6]. Lemma 4.6. For every v 4 and v 6= 6, there exists a PBD B(K, 1, v) where K = f4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 18, 19, 23, 26, 27, 30g. From Lemma 4.6 we obtain the following lemma about the existence of Mendelsohn frames with type 3 u. Lemma 4.7. A Mendelsohn frame of type 3 u exists when u 4 and u 6= 6. Proof From Lemmas 4.2 to 4.5 we know that a Mendelsohn frames of type 3 u exists for u 2 K. From Lemma 4.6 a B(K,1,u) exists for every u satisfying the hypothesis of this lemma. Give every element of the PBD wight 3 to obtain a Mendelsohn frame of type 3 u by Lemma 2.2. Lemma 4.8. If t 3 (mod 6) and u 4, u 6= 6, then there exists a Mendelsohn frame of type t u. Proof There exists a Mendelsohn frame of type 3 u by Lemma 4.7 and a RTD(3,t/3), so the conclusion follows from Lemma 2.3. At last we consider the Mendelsohn frames of type (6m + 3) 6. Lemma 4.9. There exists a Mendelsohn frame of type (6m + 3) 6 for m 62 f0, 1, 2, 3, 4, 6, 10, 14, 18, 22g. 13

14 Proof We have TD(6, m) see [1]. Give every element of one block of the TD weight 9 and other elements of the TD weight 6. As Mendelsohn frames of type 6 6 ; 9 6 ; exist (see Appendix B), we can obtain a Mendelsohn frame of type (6m + 3) 6 by Lemma 2.2. Lemma There exists a Mendelsohn frame of type t 6 for t 2 f9, 27, 63, 135g. Proof A Mendelsohn frame of type 9 6 is constructed in Appendix B and from which we can obtain Mendelsohn frames of type 27 6 ; 63 6 ; and by Lemma 2.3, since RTD(3, n) exist for n 2 f3, 7, 15g. Lemma There exists a Mendelsohn frame of type t 6 for t 2 f15, 39, 87, 111g. Proof A Mendelsohn frame of type 15 6 is constructed in Appendix B. For m = 7 or 9, there exists a TD(6; m). Give every element of one block of the TD weight 15 and other elements of the TD weight 12. As Mendelsohn frames of type 12 6 ; 15 6 ; exist (see Appendx B), we can obtain a Mendelsohn frame of type (12m + 3) 6 by Lemma 2.2. For a Mendelsohn frame of type 39 6, start with an RTD(6, 7) and delete one block. In the resulting GDD with block sizes 5 and 6 and type 6 6, we give the points of one block of size 6 weight 9, and give all the remaining points of the GDD weight 6. For input we need Mendelsohn frames of type 6 6 ; 9 6 ; 6 5 ; ; , which exist. Note the last 3 types come from GDDs with block size 4 (of the same types) which are constructed in [11]. Theorem A Mendelsohn frame of type t u exists if and only if u 4 and t(u?1) 0 (mod 3) with possible exceptions for u = 6 and t 2 f3, 21g. 14

15 Proof In view of Lemma 2.4 we need only consider the case of t 3 (mod 6). So the conclusion follows from Lemmas 4.8, 4.9, 4.10 and References [1] R.J. Abel, A.E. Brouwer, C.J. Colbourn and J.H. Dinitz, Mutually orthogonal Latin squares (MOLS), in: \The CRC Handbook of Combinatorial Designs " (C.J. Colbourn and J.H. Dinitz eds), CRC Press, New York, 1996, [2] A.M. Assaf and A. Hartman, Resolvable Group Divisible Designs with Block Size 3, Discrete Math. 77(1989), [3] F.E. Bennett and R. Wei, Embeddings of Resolvable Mendelsohn Triple Systems, J. Combinatorial Designs, 1(1993), [4] F.E. Bennett and L. Zhu, Perfect Mendelsohn Designs with Equal-Sized Holes, JCMCC, 8(1990), [5] J.C. Bermond, A. Germa and D. Sotteau, Resolvable Decomposition of Kn, J.Comb. Theory A, 26(1979), [6] T. Beth, D. Jungnickel and H. Lenz, \ Design Theory", Cambridge University Press, Cambridge, [7] A.E. Brouwer, A. Schrijver and H. Hanani, Group Divisible Designs with Block Size 4, Discrete Math. 20(1977), [8] C.J. Colbourn and A. Rosa, Directed and Mendelsohn Triple Systems in: \Contemporary Design Theory: A Collection of Surveys" (D.R. Stinson and J.H. Dinitz eds), Wiley, 1992,

16 [9] H. Hanani, Balanced Incomplete Block Designs and Related Designs, Discrete Math. 11(1975), [10] R. Rees, Two New Direct Product-type Constructions for Resolvable Group-divisible Designs, J. Combinatorial Designs, 1(1993), [11] R. Rees and D.R. Stinson, On Resolvable Group-divisible Designs with Block Size 3, Ars Combinatoria 23(1987), Appendix A. Constructions of RHMTSs RHMTS of type 3 4 Points: Z 3 Z 3 [ f1 i j1 i 3g Holes: ffig Z 3 ; ji 2 Z 3 g [ f1 i j1 i 3g Parallel classes: ((0; 0); (1; 0); (2; 1))(1 1 ; (1; 2); (0; 1))(1 2 ; (0; 2); (2; 2)) (1 3 ; (2; 0); (1; 1)) mod(3; 3) RHMTS of type 3 8 Points: Z 21 [ f1 i j1 i 3g Holes: f f i,i+7,i+14g j0 i 6g [ f1 i j1 i 3g Parallel classes: (1; 2; 20)(7; 6; 16)(19; 17; 13)(11; 3; 8)(0; 15; 5)(1 1 ; 9; 18) (1 2 ; 4; 12)(1 3 ; 10; 14) mod 21 RHMTS of type

17 Points: Z 27 [ f1 i j1 i 3g Holes: f f i,i+9,i+18g j0 i 8g [ f1 i j1 i 3g Parallel classes: (1; 2; 13)(9; 23; 20)(6; 5; 25)(15; 22; 17)(19; 11; 16)(24; 26; 12) (4; 14; 10)(1 1 ; 18; 8)(1 2 ; 21; 0)(1 3 ; 3; 7) mod 27 RHMTS of type 3 14 Points: Z 39 [ f1 i j1 i 3g Holes: f f i,i+13,i+26 g j0 i 12g [ f1 i j1 i 3g Parallel classes: (0; 29; 23)(18; 19; 15)(20; 30; 38)(34; 17; 28)(22; 7; 24)(21; 12; 9) (13; 32; 27)(35; 16; 31)(37; 25; 14)(2; 4; 36)(26; 33; 8)(1 1 ; 1; 10) (1 2 ; 11; 3)(1 3 ; 6; 5) mod 39 RHMTS of type 5 6 Points: Z 25 [ f1 i j1 i 5g Holes: f f i,i+5,i+10,i+15,i+20 g j0 i 4g [ f1 i j1 i 5g Parallel classes: (0; 16; 19)(1; 12; 13)(4; 3; 11)(22; 18; 10)(6; 8; 17)(1 1 ; 5; 2) (1 2 ; 7; 14)(1 3 ; 20; 24)(1 4 ; 23; 21)(1 5 ; 15; 9) mod 25 RHMTS of type 7 6 Points: Z 35 [ f1 i j1 i 7g Holes: f f i,i+5,i+10,i+15,i+20,i+25,i+30 g j0 i 4g [ f1 i j1 i 7g Parallel classes: (16; 20; 23)(1; 2; 13)(3; 11; 32)(22; 0; 31)(6; 8; 17)(25; 4; 33) (24; 7; 5)(1 1 ; 14; 30)(1 2 ; 18; 15)(1 3 ; 19; 26)(1 4 ; 29; 28)(1 5 ; 9; 21) (1 6 ; 10; 27)(1 7 ; 12; 34) mod 35 17

18 RHMTS of type 11 6 Points: Z 55 [ f1 i j1 i 11g Holes: f f i,i+5,i+10,,i+50 g j0 i 4g [ f1 i j1 i 11g Parallel classes: (6; 27; 33)(15; 4; 17)(18; 12; 44)(20; 31; 39)(35; 42; 51)(30; 47; 16) (3; 21; 24)(36; 32; 54)(52; 43; 26)(53; 25; 22)(7; 0; 19)(1 1 ; 23; 1) (1 2 ; 11; 34)(1 3 ; 45; 2)(1 4 ; 40; 41)(1 5 ; 49; 48)(1 6 ; 5; 9)(1 7 ; 13; 29) (1 8 ; 50; 37)(1 9 ; 46; 38)(1 10 ; 8; 10)(1 11 ; 28; 14) mod 55 RHMTS of type 13 6 Points: Z 65 [ f1 i j1 i 13g Holes: f f i,i+5,i+10,,i+60 g j0 i 4g [ f1 i j1 i 13g Parallel classes: (0; 11; 48)(56; 27; 34)(55; 42; 23)(16; 18; 60)(51; 43; 37)(20; 2; 29) (35; 32; 6)(14; 38; 21)(64; 40; 46)(39; 8; 31)(50; 49; 12)(19; 15; 3) (52; 41; 9)(1 1 ; 30; 61)(1 2 ; 25; 4)(1 3 ; 47; 1)(1 4 ; 44; 57)(1 5 ; 33; 45) (1 6 ; 36; 22)(1 7 ; 5; 54)(1 8 ; 53; 62)(1 9 ; 63; 24)(1 10 ; 58; 59)(1 11 ; 28; 26) (1 12 ; 7; 10)(1 13 ; 13; 17) mod 65 B. Constructions of Mendelsohn frames Here we display some constructions of Mendelsohn frames. We distinguish the base blocks into two groups. Every block of group I can be developed to three partial parallel classes which do not contain innite elements, since the three elements of the block are congruent to 0, 1 and 2 modulo 3 respectively. The blocks of group II form one partial parallel class and they can be developed to form partial parallel classes. Mendelsohn frame of type

19 Points: Z 21 [ f1 i j1 i 6g Holes: f f i,i+7,i+14g j0 i 6g [ f1 i j1 i 6g Blocks I: (1; 5; 0)(1; 0; 5) Blocks II: (1; 9; 11)(2; 12; 4)(1 1 ; 3; 6)(1 2 ; 10; 16)(1 3 ; 17; 5)(1 4 ; 8; 20) (1 5 ; 19; 13)(1 6 ; 18; 15) mod 21 Mendelsohn frame of type Points: Z 30 [ f1 i j1 i 12g Holes: f f i,i+10,i+20g j0 i 9g [ f1 i j1 i 12g Blocks I: (0; 2; 13)(0; 13; 2)(0; 1; 5)(0; 5; 1) Blocks II: (1; 9; 16)(1 1 ; 28; 25)(1 2 ; 13; 4)(1 3 ; 18; 11)(1 4 ; 6; 15)(1 5 ; 17; 23) (1 6 ; 7; 21)(1 7 ; 29; 2)(1 8 ; 3; 27)(1 9 ; 19; 5)(1 10 ; 12; 24)(1 11 ; 22; 14) (1 12 ; 8; 26) mod 30 Mendelsohn frame of type Points: Z 33 [ f1 i j1 i 12g Holes: f f i,i+11,i+22g j0 i 10g [ f1 i j1 i 12g Blocks I: (0; 2; 16)(0; 16; 2)(1; 5; 0)(1; 0; 5) Blocks II: (1; 9; 16)(4; 17; 7)(1 1 ; 20; 30)(1 2 ; 27; 21)(1 3 ; 2; 14)(1 4 ; 23; 26) (1 5 ; 8; 32)(1 6 ; 13; 28)(1 7 ; 5; 25)(1 8 ; 6; 12)(1 9 ; 15; 24)(1 10 ; 31; 19) (1 11 ; 3; 29)(1 12 ; 18; 10) mod 33 19

20 Mendelsohn frame of type Points: Z 39 [ f1 i j1 i 15g Holes: f f i,i+13,i+26g j0 i 12g [ f1 i j1 i 15g Blocks I: (1; 5; 0)(1; 0; 5)(0; 2; 16)(0; 16; 2)(0; 10; 17) Blocks II: (15; 24; 36)(7; 35; 4)(1 1 ; 10; 16)(1 2 ; 1; 22)(1 3 ; 38; 19)(1 4 ; 32; 25) (1 5 ; 33; 11)(1 6 ; 9; 28)(1 7 ; 5; 29)(1 8 ; 31; 3)(1 9 ; 30; 18)(1 10 ; 23; 34) (1 11 ; 17; 14)(1 12 ; 20; 12)(1 13 ; 8; 2)(1 14 ; 37; 27)(1 15 ; 6; 21) mod 39 Mendelsohn frame of type Points: Z 54 [ f1 i j1 i 24g Holes: f f i,i+18,i+36g j0 i 17g [ f1 i j1 i 24g Blocks I: (0; 2; 16)(0; 16; 2)(1; 5; 0)(1; 0; 5)(0; 10; 17)(0; 17; 10) (0; 13; 32)(0; 32; 13) Blocks II: (26; 23; 3)(1 1 ; 7; 35)(1 2 ; 4; 33)(1 3 ; 21; 51)(1 4 ; 11; 42)(1 5 ; 8; 41) (1 6 ; 32; 17)(1 7 ; 1; 43)(1 8 ; 10; 53)(1 9 ; 47; 38)(1 10 ; 24; 16)(1 11 ; 19; 13) (1 12 ; 2; 5)(1 13 ; 22; 28)(1 14 ; 52; 6)(1 15 ; 31; 40)(1 16 ; 34; 45)(1 17 ; 37; 49) (1 18 ; 15; 30)(1 19 ; 9; 29)(1 20 ; 27; 48)(1 21 ; 44; 14)(1 22 ; 25; 50)(1 23 ; 20; 46) (1 24 ; 12; 39) mod 54 Mendelsohn frame of type Points: Z 30 [ f1 i j1 i 9g Holes: f f i,i+5,i+10,i+15,i+20,i+25 g j0 i 4g [ f1 i j1 i 9g Blocks I: (0; 2; 13)(1; 0; 8)(0; 4; 23) 20

21 Blocks II: (3; 6; 12)(28; 22; 19)(1 1 ; 27; 13)(1 2 ; 7; 21)(1 3 ; 16; 29)(1 4 ; 8; 4) (1 5 ; 23; 11)(1 6 ; 2; 14)(1 7 ; 17; 18)(1 8 ; 9; 1)(1 9 ; 26; 24) mod 30 Mendelsohn frame of type 9 6 Points: Z 45 [ f1 i j1 i 9g Holes: f f i,i+5,i+10,,i+40 g j0 i 4g [ f1 i j1 i 9g Blocks I: (0; 1; 14)(0; 22; 26)(0; 41; 28) Blocks II: (3; 6; 12)(23; 44; 26)(9; 7; 16)(42; 36; 28)(41; 19; 8)(21; 39; 22) (1 1 ; 38; 17)(1 2 ; 27; 11)(1 3 ; 33; 14)(1 4 ; 1; 13)(1 5 ; 24; 31)(1 6 ; 29; 37) (1 7 ; 32; 43)(1 8 ; 2; 4)(1 9 ; 18; 34) mod 45 Mendelsohn frame of type 15 6 Points: Z 75 [ f1 i j1 i 15g Holes: f f i,i+5,i+10,,i+70 g j0 i 4g [ f1 i j1 i 15g Blocks I: (0; 1; 8)(0; 28; 11)(0; 59; 37)(0; 17; 64)(0; 31; 29) Blocks II: (64; 22; 8)(24; 21; 73)(68; 32; 19)(59; 53; 2)(28; 1; 67)(74; 62; 6) (26; 44; 58)(13; 57; 56)(23; 27; 61)(14; 36; 12)(1 1 ; 4; 16)(1 2 ; 7; 34) (1 3 ; 51; 18)(1 4 ; 48; 69)(1 5 ; 3; 11)(1 6 ; 17; 71)(1 7 ; 33; 39)(1 8 ; 41; 37) (1 9 ; 49; 52)(1 10 ; 47; 63)(1 11 ; 29; 42)(1 12 ; 31; 72)(1 13 ; 43; 66)(1 14 ; 54; 46) (1 15 ; 9; 38) mod 75 Mendelsohn frame of type Points: Z 60 [ f1 i j1 i 15g 21

22 Holes: f f i,i+5,i+10,,i+55, g j0 i 4g [ f1 i j1 i 15g Blocks I: (0; 2; 28)(0; 7; 26)(0; 11; 4)(0; 17; 1)(0; 4; 17) Blocks II: (6; 14; 17)(2; 24; 33)(56; 42; 23)(29; 27; 51)(34; 46; 37)(58; 21; 22) (1 1 ; 26; 54)(1 2 ; 52; 31)(1 3 ; 32; 48)(1 4 ; 47; 39)(1 5 ; 44; 38)(1 6 ; 1; 19) (1 7 ; 16; 3)(1 8 ; 4; 18)(1 9 ; 7; 13)(1 10 ; 41; 8)(1 11 ; 12; 43)(1 12 ; 59; 36) (1 13 ; 9; 57)(1 14 ; 11; 53)(1 15 ; 28; 49) mod 60 22

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