ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.1056547 J. Algebra Comb. Discrete Appl. 9(1) • 29–46 Received: 24 January 2021 Accepted: 25 November 2021 Journal of Algebra Combinatorics Discrete Structures and Applications Decomposition of cartesian product of complete graphs into sunlet graphs of order eight∗ Research Article Kaliappan Sowndhariya, Appu Muthusamy Abstract: For any integer k ≥ 3, we define the sunlet graph of order 2k, denoted by L2k, as the graph consisting of a cycle of length k together with k pendant vertices such that, each pendant vertex adjacent to exactly one vertex of the cycle so that the degree of each vertex in the cycle is 3. In this paper, we establish necessary and sufficient conditions for the existence of decomposition of the Cartesian product of complete graphs into sunlet graphs of order eight. 2010 MSC: 05C51 Keywords: Graph decomposition, Cartesian product, Corona graph, Sunlet graph 1. Introduction All graphs considered here are finite, simple and undirected. A cycle of length k is called k-cycle and it is denoted by Ck. Km denotes the complete graph on m vertices and Km,n denotes the complete bipartite graph with m and n vertices in the parts. We denote the complete m-partite graph with n1,n2, . . . ,nm vertices in the parts by Kn1,n2,...,nm. For any integer λ > 0, λG denotes the graph consisting of λ edge-disjoint copies of G. Let G and H be two graphs of orders m and n, respectively. The corona prod- uct G � H is the graph obtained by taking one copy of G and m copies of H such that the ith vertex of G is connected to every vertex in the ith copy of H. We de- fine the sunlet graph L2k with V (L2k) = {x1,x2, . . . ,xk,xk+1,xk+2, . . . ,x2k} and E(L2k) = {xixi+1 ∪xixk+i | i = 1,2, ...,k and subscripts of the first term is taken addition modulo k}. We denote it by L2k = ( x1 x2 . . . xk xk+1 xk+2 . . . x2k ) . Clearly, L2k ∼= Ck �K1. ∗ This work was supported by Department of Science and Technology, University Grant Commission, Government of India. Kaliappan Sowndhariya, Appu Muthusamy (Corresponding Author); Department of Mathematics, Periyar Uni- versity, Salem, Tamil Nadu, India (email: sowndhariyak@gmail.com, ambdu@yahoo.com). 29 https://orcid.org/0000-0001-9014-6916 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 The Cartesian product of two graphs, G and H, denoted by G�H, has the vertex set V (G)×V (H) and two vertices (g,h) and (g′,h′) are adjacent if and only if either g = g′ and h is adjacent to h′ in H or h = h′ and g is adjacent to g′ in G. It is well known that Cartesian product is commutative, associative and distributive over edge-disjoint union of graphs. We shall use the following notation throughout the paper. Let G and H be simple graphs with vertex sets V (G) = {x1,x2, . . . ,xn} and V (H) = {y1,y2, . . . ,ym}. Then for our convenience, we write V (G) × V (H) = ⋃n i=1 Xi, where Xi stands for xi × V (H). Further, in the sequel, we shall denote the vertices of Xi as { x j i|1 ≤ j ≤ m } , where xji stands for the vertex (xi,yj) ∈ V (G)×V (H). By a decomposition of a graph G, we mean a list of edge-disjoint subgraphs whose union is G. For a graph G, if E(G) can be partitioned into E1,E2, ...,Ek such that the subgraph induced by Ei is Hi, for all i, 1 ≤ i ≤ k, then we say that H1,H2, ...,Hk decompose G and we write G = H1 ⊕ H2 ⊕ ... ⊕ Hk, since H1,H2, ...,Hk are edge-disjoint subgraphs of G. For 1 ≤ i ≤ k, if Hi = H, we say that G has a H- decomposition. Study of H-decomposition of graphs is not new. Many authors around the world are working in the field of cycle decomposition [4, 7, 8, 18, 19], path decomposition [22, 23], star decompositon [17, 21, 24, 25] and Hamilton cycle decomposition [2, 3, 13, 14] problems in graphs. Here we consider the sunlet decomposition of product graphs. Anitha and Lekshmi [5, 6] proved that n-sun decomposition of complete graph, complete bipartite graph and the Harary graphs. Liang and Guo [15, 16] gave the existence spectrum of a k-sun system of order v as k = 2,4,5,6,8. Fu et. al. [10, 11] obtained that 3-sun decompositions of Kp,p,r, Knand embed a cyclic steiner triple system of order n into a 3-sun system of order 2n − 1, for n ≡ 1 (mod 6). Further they obtained k-sun system when k = 6,10,14,2t, for t > 1. Fu et. al. [9] obtained the existence of a 5-sun system of order v. Gionfriddo et.al. [12] obtained the spectrum for uniformly resolvable decompositions of Kv into 1-factor and h-suns. Akwu and Ajayi [1] obtained the necessary and sufficient conditions for the existence of decomposition of Kn ⊗ Km and (Kn − I) ⊗ Km, where I denote the 1-factor of a complete graph into sunlet graph of order 2p, p is a prime. Sowndhariya and Muthusamy [20] obtained necessary and sufficient conditions for the existence of decomposition of Km ×Kn and Km ⊗Kn into sunlet graph of order eight. In this paper, we prove the existence of an L8-decomposition of Km�Kn. In fact, we establish necessary and sufficient conditions for the existence of an L8-decomposition of Km�Kn. To prove our results, we state the following: Theorem 1.1. [11] Let t ≥ 2 be an integer. An L2.2t -decomposition of Kn exists if and only if n ≡ 0 (or) 1 (mod 2t+2). Theorem 1.2. [20] For any m,n ≥ 4, Km,n has an L8- decomposition if and only if mn ≡ 0 (mod 8) except (m,n) = (4,2 (mod 4)) & (8,5). 2. Decomposition of Km�Kn into sunlet graph of order 8 Necessary conditions: Lemma 2.1. If Km�Kn has an L8-decomposition, then either 1. m,n ≡ 0 (mod 4) 2. m ≡ 0 (mod 8), n ≡ 0 (mod 2) 3. m ≡ 4 (mod 8), n ≡ 2 (mod 4) 4. m ≡ 0 (mod 16) 5. m ≡ 1 (mod 16), n ≡ 1 (mod 16) 6. m ≡ 15 (mod 16), n ≡ 3 (mod 16) 30 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 7. m ≡ 13 (mod 16), n ≡ 5 (mod 16) 8. m ≡ 11 (mod 16), n ≡ 7 (mod 16) 9. m ≡ 9 (mod 16), n ≡ 9 (mod 16) Proof. The graph Km�Kn has mn vertices, each having degree m + n−2 and hence has mn(m+n−2) 2 edges. Assume that Km�Kn admits an L8- decomposition. Then the number of edges in the graph must be divisible by 8. i.e., 16|mn(m + n−2). Hence these conditions are met in each of the above nine cases and only in these cases. Figure 1. L8-decomposition of K4�K2. Sufficient conditions: We now prove the above necessary conditions are also sufficient by proving the following Lemmas: Lemma 2.2. If m ≡ 0 (mod 4) and n ≡ 0 (mod 4), then the graph Km�Kn has an L8-decomposition. Proof. Let m = 4s and n = 4t for some s,t > 0. We can divide the graph Km�Kn into st(K4�K4), the L8-decomposition of K4�K4 is shown in Fig. 2 and the remaining edges are viewed in the following manner; for each row we have a ’t’ set of four vertices and each set is adjacent to other. Therefore we get t(t − 1)/2 complete bipartite graph K4,4. Then by Theorem 1.2, K4,4 has an L8-decomposititon. Similarly we can use the same procedure to column vertices. Hence the graph Km�Kn has the desired decomposition. Lemma 2.3. If m ≡ 0 (mod 8) and n ≡ 0 (mod 2), then the graph Km�Kn has an L8-decomposition. Proof. For m = 8 and n = 2, the graph K8�K2 has an L8- decomposition by Fig. 1 and by Theorem 1.2. Let m ≡ 0 (mod 8) and n ≡ 0 (or) 4 (mod 8), then the proof follows from Lemma 2.2. Set m ≡ 0 (mod 8) and consider two cases for n. Case (1) n ≡ 2 (mod 8). Let m = 8s and n = 8t + 2 for some s,t > 0. The graph Km�Kn can be viewed as s(t−1) (K8�K8) ⊕ s(K8�K10) and the remaining edges viewed as follows; in each row, we have (t−1) set of eight vertices and one set of ten vertices which are form the complete bipartite graph K8,8 and K8,10. Similarly each column can be viewed as ’s’ set of eight vertices and each set is adjacent with each other (i.e. K8,8). The L8-decomposition of K8�K10 is given in Appendix 3.1.1 and the L8-decomposition of the graphs K8�K8, K8,8 and K8,10 follows from Lemma 2.2 and Theorem 1.2. Case (2) n ≡ 6 (mod 8). Let m = 8s and n = 8t+6 for some s,t > 0. Then we can view Km�Kn as st(K8�K8) ⊕ s(K8�K6) and the remaining edges form the complete bipartite graph K8,8 and K8,6 which are obtained by using the above procedure. The L8-decomposition of K8�K6 is given in Appendix 3.1.2 and the L8-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.2. Hence the graph Km�Kn has the desired decomposition. Lemma 2.4. If m ≡ 4 (mod 8) and n ≡ 2 (mod 4), then the graph Km�Kn has an L8-decomposition. 31 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 Figure 2. L8-decomposition of K4�K4. Proof. Let m = 8s + 4 and n = 4t + 2 for some s,t > 0. Now, we divide the proof into three cases. Case (1) m = 4 and n = 4t + 2 for some t > 0. If t = 1, then the graph K4�K6 has an L8-decomposition, see Appendix 3.2.1. If t = 2, then the graph K4�K10 has an L8-decomposition, see Appendix 3.2.2. Further, for t > 2, the graph K4�K4t+2 can be viewed as K4�K6 ⊕ K4�K4(t−1) ⊕ 4K6,4(t−1). Then by Appendix 3.2.1, Lemma 2.2 and Theorem 1.2, we get the desired decomposition. Case (2) m = 12 and n = 4t + 2 for some t > 0. If t = 1, then the graph K12�K6 has an L8-decomposition see Appendix 3.2.3. If t = 2, then the graph K12�K10 has an L8-decomposition see Appendix 3.2.4. Further, for t > 2, the graph K12�K4t+2 can be viewed as K12�K6 ⊕ K12�K4(t−1) ⊕ 12K6,4(t−1). Then by Appendix 3.2.3, Lemma 2.2 and Theorem 1.2, we get the desired decomposition. Case (3) m > 12 and n = 4t + 2 for some t > 0. The graph Km�Kn can be viewed as K8(s−1)�K4t+2 ⊕ K12�K4t+2 ⊕ (4t + 2)K12,8(s−1). Then by Lemma 2.3, Theorem 1.2 and the above Case (2), we get the desired decomposition. Lemma 2.5. If m ≡ 0 (mod 16), then the graph Km�Kn has an L8-decomposition. Proof. If n = 1, the graph Km has an L8-decomposition by Theorem 1.1. Let m ≡ 0 (mod 16) and n ≡ 0,2,4,6 (mod 8), then the proof follows from Lemma 2.3. Set m = 0 (mod 16) and consider four cases for odd n. Case (1) n ≡ 1 (mod 8). Let m = 16s and n = 8t + 1 for some s,t > 0. The graph Km�Kn can be viewed as s(t− 1) (K16�K8) ⊕ s(K16�K9). Then the remaining edges are viewed as follows; each row contains (t − 1) set of eight vertices and one set of nine vertices. Then each set is adjacent to each other and these forms the complete bipartite graphs K8,8, K8,9. Similarly each column can be viewed as ’s’ set of sixteen vertices and each set is adjacent with each other (i.e. we have K16,16). Finally theL8-decomposition of K16�K9 is shown in Appendix 3.3.1 and the L8-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.2. Case (2) n ≡ 3 (mod 8). If n = 3, then the graph Km�K3 can be viewed as copies of K16�K3 and K16,16 which has an L8- decomposition, see Appendix 3.3.2 and Theorem 1.2. Let m = 16s and n = 8t + 3 for some s,t > 0. 32 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 Apply the same procedure as in Case (1) and we write Km�Kn as s(t−1) (K16�K8) ⊕ s(K16�K11) ⊕ 8s(t−1)(t−2)K8,8 ⊕ s(8t+3)(s−1) 2 K16,16 ⊕ 16s(t−1)K8,11. An L8-decomposition of K16�K11 is shown in Appendix 3.3.3 and the L8-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.2. Case (3) n ≡ 5 (mod 8). If n = 5, then the graph Km�K5 can be viewed as K16�K5 ⊕ K16,16 which has an L8-decomposition, see Appendix 3.3.4 and Theorem 1.2. Let m = 16s and n = 8t + 5 for some s,t > 0. Then we can write Km�Kn as s(t − 1) (K16�K8) ⊕ s(K16�K13) ⊕ 8s(t − 1)(t − 2)K8,8 ⊕ s(8t+5)(s−1) 2 K16,16 ⊕ 16s(t − 1)K8,13. An L8-decomposition of K16�K13 is presented in Appendix 3.3.5 and the L8-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.2. Case (4) n ≡ 7 (mod 8). Let m = 16s and n = 8t+7 for some s,t > 0. Then we can write Km�Kn as st(K16�K8) ⊕ s(K16�K7) ⊕ 8st(t − 1)K8,8 ⊕ s(8t+7)(s−1) 2 K16,16 ⊕ 16stK8,7. An L8-decomposition of K16�K7 is presented in Appendix 3.3.6 and the L8-decomposition of the remaining graphs follows from Lemma 2.2 and Theorem 1.2. Hence the graph Km�Kn has the desired decomposition. Lemma 2.6. If m ≡ 1 (mod 16) and n ≡ 1 (mod 16), then the graph Km�Kn has an L8-decomposition. Proof. Let m = 16s + 1 and n = 16t + 1 for some s,t > 0. Then we can write Km�Kn = nKm ⊕ mKn. i.e, (16t + 1)K16s+1 ⊕ (16s + 1)K16t+1. By Theorem 1.1, the graph Km�Kn has the desired decomposition. Lemma 2.7. If m ≡ 15 (mod 16) and n ≡ 3 (mod 16), then the graph Km�Kn has an L8- decomposition. Proof. Let m = 16s+ 15 and n = 16t+ 3 for some s,t > 0. We can write Km�Kn as (16t+ 3)K16s+15 ⊕ (16s + 15)K16t+3. Now the first 16t columns can be viewed as K16s ⊕ K15 ⊕ sK16,15 and the first 16s rows can be viewed as K16(t−1) ⊕ K19 ⊕ (t − 1)K16,19. Then K16s(= sK16 ⊕ s(s−1) 2 K16,16), K16(t−1)(= (t− 1)K16 ⊕ (t−1)(t−2) 2 K16,16), K16,15 and K16,19 have L8-decompositions by Theorems 1.1, 1.2. The graph K19 can be viewed as K19\K3 ⊕ K3. The L8-decomposition of K19\K3 follows from Appendix 3.4.1. Then 16s(K19\K3) has an L8-decomposition. The remaining graph can be viewed as s(K16�K3) ⊕ t(K15�K16) ⊕ K15�K3. Hence the desired decomposition follows from Appendixes 3.3.2, 3.4.2 and Lemma 2.5 Lemma 2.8. If m ≡ 13 (mod 16) and n ≡ 5 (mod 16), then the graph Km�Kn has an L8- decomposition. Proof. Let m = 16s+13 and n = 16t+5 for some s,t > 0. Then we can write Km�Kn as st(K16�K16) ⊕ t(K16�K13) ⊕ s(K16�K5) ⊕ K13�K5 ⊕ t(t−1)(16s+13)+s(s−1)(16t+5) 2 K16,16 ⊕ s(16t + 5)K16,13 ⊕ t(16s+13)K16,5. An L8-decomposition of K13�K5 is given in Appendix 3.5.1. and the L8-decomposition of the remaining graphs follows from Lemma 2.5 and Theorem 1.2. Hence the graph Km�Kn has the desired decomposition. Lemma 2.9. If m ≡ 11 (mod 16) and n ≡ 7 (mod 16), then the graph Km�Kn has an L8- decomposition. Proof. Let m = 16s+11 and n = 16t+7 for some s,t > 0. Then we can write Km�Kn as st(K16�K16) ⊕ t(K16�K11) ⊕ s(K16�K7) ⊕ K11�K7 ⊕ s(s−1)(16t+7)+t(t−1)(16s+11) 2 K16,16 ⊕ (16t + 7)sK16,11 ⊕ (16s+ 11)tK16,7. An L8-decomposition of K11�K7 is given in Appendix 3.6.1 and the L8-decomposition of the remaining graphs follows from Lemma 2.5 and Theorem 1.2. Hence the graph Km�Kn has the desired decomposition. Lemma 2.10. If m ≡ 9 (mod 16) and n ≡ 9 (mod 16), then the graph Km�Kn has an L8- decomposition. 33 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 Proof. Let m = 16s+9 and n = 16t+9 for some s,t > 0. Then we can write Km�Kn as st(K16�K16) ⊕ (s + t) (K16�K9) ⊕ K9�K9 ⊕ s(s−1)(16t+9)+t(t−1)(16s+9) 2 K16,16 ⊕ [s(16t + 9) + t(16s + 9)]K16,9. An L8-decomposition of K11�K7 is given in Appendix 3.7.1 and the L8-decomposition of the remaining graphs follows from Lemma 2.5 and Theorem 1.2. Hence the graph Km�Kn has the desired decomposi- tion. 2.1. Main theorem Combining the results from Lemma 2.1 to Lemma 2.10, we get the following main result. Theorem 2.11. The graph Km�Kn admits an L8- decomposition if and only if one of the following holds: 1. m,n ≡ 0 (mod 4) 2. m ≡ 0 (mod 8), n ≡ 0 (mod 2) 3. m ≡ 4 (mod 8), n ≡ 2 (mod 4) 4. m ≡ 0 (mod 16) 5. m ≡ 1 (mod 16), n ≡ 1 (mod 16) 6. m ≡ 15 (mod 16), n ≡ 3 (mod 16) 7. m ≡ 13 (mod 16), n ≡ 5 (mod 16) 8. m ≡ 11 (mod 16), n ≡ 7 (mod 16) 9. m ≡ 9 (mod 16), n ≡ 9 (mod 16) Acknowledgment: The first author thank the Department of Science and Technology, Gov- ernment of India, New Delhi for its financial support through the Grant No.DST/INSPIRE Fellow- ship/2015/IF150211. The second author thank the University Grant Commission, Government of India, New Delhi (Grant No. F.510/7/DRS-I/2016(SAP-DRS-I)) and the Department of Science and Technol- ogy, New Delhi (Grant No. SR/FIST/MSI-115/2016(Level-I)), for their generous financial support. References [1] A. D. Akwu, D. O. A. Ajayi, Decomposing certain equipartite graphs into sunlet graphs of length 2p, AKCE Int. J. Graphs Combin. 13(3) (2016) 267–271. 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Yamamoto, On claw-decomposition of complete multipartite graphs, Hi- roshima Math. J. 8(1) (1978) 207–210. [25] S. Yamamoto, H. Ikeda, S. Shige-Eda, K. Ushio, N. Hamada, On claw decomposition of complete graphs and complete bipartite graphs, Hiroshima Math. J. 5(1) (1975) 33-42. 3. Appendix 3.1. L8-decomposition required for Lemma 2.3 3.1.1. An L8- decomposition of K8�K10( x1i x 6 i x 2 i x 7 i x3i x 8 i x 4 i x 9 i ) , ( x3i x 8 i x 4 i x 9 i x10i x 5 i x 7 i x 2 i ) , ( x2i x 3 i x 4 i x 5 i x1i x 7 i x 6 i x 9 i ) for i = 1, 2, ..., 8;( x j 1 x j 5 x j 2 x j 6 x j 3 x j 7 x j 4 x j 8 ) , ( x j 3 x j 7 x j 4 x j 8 x j 5 x j 1 x j 6 x j 2 ) for j = 1, 2, ..., 10;( x j 1 x j 2 x j 3 x j 4 x j 8 x j 7 x j 6 x j 5 ) for j = 1, 3, 5, 7, 8, 9; ( x j 5 x j 6 x j 7 x j 8 x j 4 x j 3 x j 2 x j 1 ) for j = 2, 4, 6, 10;( x1i x 5 i x 6 i x 10 i x8i x k1 i x k2 i x k3 i ) for i = 1, 2, 3, 4, (k1, k2, k3) = (3, 9, 4) & i = 5, 6, 7, 8, (k1, k2, k3) = (7, 3, 2);( x7i x 8 i x 9 i x 10 i x6i x 2 i x 1 i x k i ) for i = 1, 2, 3, 4, k = 5 & i = 5, 6, 7, 8, k = 4; 35 https://doi.org/10.1201/9781420010541 https://doi.org/10.1201/9781420010541 https://doi.org/10.1016/j.disc.2013.09.007 https://doi.org/10.1016/j.disc.2013.09.007 https://doi.org/10.11650/twjm/1500406600 https://doi.org/10.11650/twjm/1500406600 https://doi.org/10.1016/j.disc.2012.03.007 https://doi.org/10.1016/j.disc.2012.03.007 https://mathscinet.ams.org/mathscinet-getitem?mr=3495890 https://mathscinet.ams.org/mathscinet-getitem?mr=3495890 https://doi.org/10.1016/0095-8956(84)90020-0 https://doi.org/10.1016/0095-8956(84)90020-0 https://doi.org/10.1016/0012-365X(86)90186-X https://doi.org/10.1016/0012-365X(86)90186-X https://doi.org/10.1007/s12190-009-0267-0 https://doi.org/10.1007/s12190-009-0267-0 https://doi.org/10.1002/(SICI)1097-0118(199612)23:4<361::AID-JGT5>3.0.CO;2-P https://doi.org/10.1002/(SICI)1097-0118(199612)23:4<361::AID-JGT5>3.0.CO;2-P https://doi.org/10.1002/jcd.1027 https://doi.org/10.1002/jcd.1027 https://doi.org/10.1016/0095-8956(81)90093-9 https://doi.org/10.1016/0095-8956(81)90093-9 https://doi.org/10.13069/jacodesmath.867617 https://doi.org/10.13069/jacodesmath.867617 https://doi.org/10.1016/0012-365X(79)90034-7 https://doi.org/10.1016/0097-3165(83)90040-7 https://doi.org/10.1016/0097-3165(83)90040-7 https://doi.org/10.1016/S0012-365X(85)80023-6 https://doi.org/10.1016/S0012-365X(85)80023-6 https://doi.org/10.32917/hmj/1206135570 https://doi.org/10.32917/hmj/1206135570 https://doi.org/10.32917/hmj/1206136782 https://doi.org/10.32917/hmj/1206136782 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 ( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) for (j, k) = (2, 10), (4, 1), (6, 3), (10, 8);( x j 5 x j 6 x j 7 x j 8 xk5 x k 6 x k 7 x k 8 ) for (j, k) = (1, 4), (3, 5), (5, 10), (7, 5), (8, 10), (9, 6). 3.1.2. An L8- decomposition of K8�K6( x j 1 x j 5 x j 2 x j 6 xk1 x j 7 x k 2 x j 8 ) , ( x j 3 x j 7 x j 4 x j 8 xk3 x j 1 x k 4 x j 2 ) for (j, k) = (1, 3), (5, 2), (6, 4);( x j 1 x j 5 x j 2 x j 6 x j 3 x k 5 x j 4 x k 6 ) ; ( x j 3 x j 7 x j 4 x j 8 x j 5 x k 7 x j 6 x k 8 ) for (j, k) = (2, 5), (3, 1), (4, 6);( x1i x 2 i x 3 i x 4 i x1k x 2 9−i x 3 9−i x 4 9−i ) for (i, k) = (1, 3), (2, 4), (3, 6);( x1i x 5 i x 3 i x 6 i x1i+1 x 5 k x 3 i+1 x 6 k ) for (i, k) = (1, 8), (2, 4), (3, 5);( x1i x 2 i x 3 i x 4 i x1i+1 x 2 i+1 x 3 i+1 x 4 i+1 ) for i = 4, 5, 6, 7; ( x2i x 4 i x 5 i x 6 i x2i+1 x 4 i+1 x 5 i+1 x 6 i+1 ) for i = 1, 2, 3;( x18 x 2 8 x 3 8 x 4 8 x15 x 2 5 x 3 5 x 4 5 ) , ( x14 x 5 4 x 3 4 x 6 4 x11 x 5 6 x 3 1 x 6 6 ) , ( x15 x 5 5 x 3 5 x 6 5 x13 x 5 6 x 3 7 x 6 6 ) , ( x16 x 5 6 x 3 6 x 6 6 x14 x 5 7 x 3 8 x 6 7 ) ,( x17 x 5 7 x 3 7 x 6 7 x12 x 5 8 x 3 1 x 6 8 ) , ( x18 x 5 8 x 3 8 x 6 8 x11 x 5 5 x 3 2 x 6 5 ) , ( x24 x 4 4 x 5 4 x 6 4 x21 x 4 1 x 5 1 x 6 1 ) , ( x25 x 4 5 x 5 5 x 6 5 x27 x 4 7 x 5 4 x 6 4 ) ,( x26 x 4 6 x 5 6 x 6 6 x28 x 4 8 x 5 3 x 6 3 ) , ( x27 x 4 7 x 5 7 x 6 7 x21 x 4 1 x 5 2 x 6 2 ) , ( x28 x 4 8 x 5 8 x 6 8 x22 x 4 2 x 5 1 x 6 1 ) . 3.2. L8-decomposition required for Lemma 2.4 3.2.1. An L8- decomposition of K4�K6( x2i x 3 i x 6 i x 4 i x2k x 3 k x 1 i x 5 i ) for (i, k) = (1, 3), (2, 4); ( x2i x 3 i x 6 i x 4 i x5i x 1 i x 6 k x 4 k ) for (i, k) = (3, 1), (4, 2);( x1i x 5 i x 5 k x 1 k x2i x 6 i x 6 k x 2 k ) for (i, k) = (1, 3), (2, 4); ( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) for (j, k) = (1, 4), (4, 3);( x j 1 x j 2 x j 3 x j 4 x k1 1 x k1 2 x k2 3 x k2 4 ) for (j, k1, k2) = (2, 5, 6), (3, 1, 5), (5, 3, 4), (6, 2, 1). 3.2.2. An L8- decomposition of K4�K10( x1i x 9 i x 2 i x 10 i x1k x 3 i x 2 k x 4 i ) , ( x7i x 9 i x 8 i x 10 i x4i x 5 i x 6 i x 10 k ) , ( x3i x 5 i x 5 k x 3 k x1i x 10 i x 2 k x 1 k ) for (i, k) = (1, 3), (2, 4);( x7i x 9 i x 8 i x 10 i x7k x 9 k x 8 k x 3 i ) for (i, k) = (3, 1), (4, 2); ( x4i x 6 i x 6 k x 4 k x5i x 1 i x 8 k x 5 k ) for (i, k) = (1, 3), (2, 4);( x1i x 2 i x 3 i x 4 i x7i x 6 i x 8 i x 9 i ) for i = 1, 2, 3, 4; ( x5i x 6 i x 7 i x 8 i x2i x 10 i x 3 i x 4 i ) for i = 1, 2;( x5i x 6 i x 7 i x 8 i x9i x 10 i x 3 i x 4 i ) , ( x1i x 9 i x 2 i x 10 i x6i x 3 i x 4 i x 5 i ) for i = 3, 4;( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) for (j, k) = (1, 5), (2, 8), (3, 6), (5, 7), (6, 9), (7, 2), (8, 1), (9, 10);( x41 x 4 2 x 4 3 x 4 4 x21 x 2 2 x 7 3 x 7 4 ) , ( x101 x 10 2 x 10 3 x 10 4 x31 x 3 2 x 4 3 x 4 4 ) . 3.2.3. An L8- decomposition of K12�K6( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x j 6 x j 5 ) , ( x j 5 x j 6 x j 7 x j 8 xk5 x k 6 x j 10 x j 9 ) , ( x j 9 x j 10 x j 11 x j 12 xk9 x k 10 x j 4 x j 3 ) for (j, k) = (1, 3), (2, 4); 36 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 ( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) , ( x j 5 x j 6 x j 7 x j 8 xk5 x k 6 x k 7 x k 48 ) , ( x j 9 x j 10 x j 11 x j 12 xk9 x k 10 x k 11 x k 12 ) for (j, k) = (3, 5), (4, 6), (5, 4), (6, 3); ( x j 5 x j 9 x j 6 x j 10 xk12 x k 2 x j 11 x j 1 ) for j = 1, ..., 6;( x j 1 x j 5 x j 2 x j 6 x j 7 x j 11 x j 8 x j 12 ) , ( x j 3 x j 9 x j 4 x j 10 x j 5 x j 1 x j 6 x j 2 ) , ( x j 7 x j 11 x j 8 x j 12 x j 9 x j 3 x j 10 x j 4 ) for j = 1, ..., 6;( x j 1 x j 11 x j 2 x j 12 x j 8 x j 9 x j 7 x j 10 ) , ( x j 3 x j 7 x j 4 x j 8 x j 1 x j 5 x j 2 x j 6 ) for j = 1, 2;( x j 1 x j 11 x j 2 x j 12 x j 8 x j 4 x j 7 x j 3 ) , ( x j 3 x j 7 x j 4 x j 8 x j 6 x j 10 x j 5 x j 9 ) for j = 5, 6;( x j 1 x j 11 x j 2 x j 12 x j 8 x k 11 x j 7 x k 12 ) , ( x j 3 x j 7 x j 4 x j 8 xk3 x k 7 x k 4 x k 8 ) for (j, k) = (3, 1), (4, 2);( x5i x 6 i x 6 k x 5 k x2i x 1 i x 1 k x 2 k ) for (i, k) = (1, 3), (2, 4), (5, 7), (6, 8), (9, 11), (10, 12);( x1i x 2 i x 3 i x 4 i x5i x 6 i x 3 k x 4 k ) for (i, k) = (1, 3), (2, 4), (3, 6), (4, 5), (5, 7), (6, 8), (7, 10), (8, 9), (9, 11), (10, 12), (11, 4), (12, 3). 3.2.4. An L8- decomposition of K12�K10( x1i x 2 i x 3 i x 4 i x1k x 2 k x 3 k x 4 k ) , ( x5i x 6 i x 7 i x 8 i x5k x 6 k x 7 k x 8 k ) , ( x1i x 9 i x 2 i x 10 i x6i x 9 k x 5 i x 10 k ) for (i, k) = (1, 7), (2, 8), (3, 6), (4, 5), (5, 11), (6, 12), (7, 10), (8, 9), (9, 2), (10, 1), (11, 3), (12, 4);( x7i x 9 i x 8 i x 10 i x7k x 9 k x 8 k x 10 k ) for (i, k) = (1, 8), (2, 7), (3, 12), (4, 11), (5, 12), (6, 11); c ( x7i x 9 i x 8 i x 10 i x3i x 5 i x 6 i x 4 i ) for i = 7, ..., 12; ( x3i x 5 i x 4 i x 6 i x7i x 10 i x 2 i x 8 i ) for i = 1, ..., 6;( x3i x 5 i x 4 i x 6 i x3k x 5 k x 4 k x 6 k ) for (i, k) = (7, 2), (8, 1), (9, 11), (10, 12), (11, 6), (12, 5);( x j 1 x j 2 x j 3 x j 4 xk1 x k 2 x k 3 x k 4 ) for (j, k) = (1, 3), (3, 10), (4, 7), (5, 7), (6, 10), (7, 1)(8, 2), (9, 3), (10, 4);( x j 5 x j 6 x j 7 x j 8 xk5 x k 6 x k 7 x k 8 ) for (j, k) = (1, 3), (3, 7), (4, 7), (5, 7), (6, 8), (7, 1)(8, 2), (9, 3);( x j 9 x j 10 x j 11 x j 12 xk9 x k 10 x k 11 x k 12 ) for (j, k) = (1, 3), (2, 4), (3, 7), (4, 7), (5, 7), (6, 8), (7, 1)(8, 2), (9, 3), (10, 5);( x j 1 x j 5 x j 2 x j 6 xi3 x k 5 x i 4 x k 6 ) for (j, k) = (3, 8), (4, 8), (5, 9), (6, 9);( x j 1 x j 5 x j 2 x j 6 xk1 x k 5 x k 2 x k 6 ) for (j, k) = (1, 5), (2, 6), (7, 2), (8, 1), (9, 4), (10, 9);( x j 3 x j 9 x j 4 x j 10 xk3 x k 9 x k 4 x k 10 ) for (j, k) = (2, 6), (3, 8), (4, 8), (6, 9), (7, 2), (8, 1), (9, 4), (10, 9);( x j 7 x j 11 x j 8 x j 12 xk7 x k 11 x k 8 x k 12 ) for (j, k) = (2, 6), (3, 8), (4, 8), (5, 1), (6, 9), (7, 2), (8, 1), (9, 4), (10, 9);( x j 1 x j 11 x j 2 x j 12 x j 3 x j 9 x j 4 x j 10 ) for i = 2, 7, 8, 9, 10;( x j 1 x j 11 x j 2 x j 12 xk1 x j 4 x k 2 x j 3 ) for (j, k) = (3, 9), (4, 8), (5, 9), (6, 9);( x j 5 x j 9 x j 6 x j 10 x j 7 x j 1 x j 8 x j 2 ) , ( x j 3 x j 7 x j 4 x j 8 x j 5 x j 9 x j 6 x j 10 ) , for j = 1, ..., 10;( x21 x 2 2 x 2 3 x 2 4 x28 x 2 7 x 2 12 x 2 11 ) , ( x25 x 2 6 x 2 7 x 2 8 x212 x 2 11 x 4 7 x 4 8 ) , ( x105 x 10 6 x 10 7 x 10 8 x45 x 4 6 x 4 7 x 4 8 ) , ( x13 x 1 9 x 1 4 x 1 10 x53 x 1 11 x 5 4 x 1 12 ) ,( x53 x 5 9 x 5 4 x 5 10 x93 x 1 9 x 9 4 x 1 10 ) , ( x17 x 1 11 x 1 8 x 1 12 x12 x 1 4 x 1 1 x 1 3 ) , ( x11 x 1 11 x 1 2 x 1 12 x13 x 1 6 x 1 4 x 1 5 ) . 37 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 3.3. L8-decomposition required for Lemma 2.5 3.3.1. An L8- decomposition of K16�K9( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x j 5 ) , ( x j 3 x j 4 x j 13 x j 14 x j 12 x j 11 x j 5 x j 15 ) for j = 1, 4, 5, 7, 8;( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x j 2 x j 16 ) , ( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x j 11 x j 1 ) for j = 1, 4, 5, 7, 8;( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) , ( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) for j = 1, 3, 4, 5, 6, 7, 8;( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) , ( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x j 3 x j 14 ) for j = 1, 3, 4, 5, 6, 7, 8, 9;( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) , ( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) for j = 1, 3, 4, 5, 7, 8;( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) for j = 1, 3, 4, 5, 6, 7, 8; ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 1, 3, 4, 5, 7, 8;( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 1, 2, 3, 4, 5, 6, 7, 8; ( x j 1 x j 5 x j 2 x j 6 x j 3 x j 16 x j 12 x j 14 ) for j = 6, 9;( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x j 16 ) for j = 1, 2, ..., 8, 9; ( x3i x 7 i x 4 i x 9 i x1i x 5 i x 6 i x 8 i ) for i = 1, 2, ..., 15, 16;( x9i x 9 i+4 x 9 i+1 x 9 i+5 x9i+2 x 5 i+4 x 9 i+3 x 5 i+5 ) for i = 5, 7; ( x1i x 8 i x 2 i x 9 i x5i x 6 i x 4 i x 7 i ) for i = 1, 2, ..., 7, 8;( x j i x j 12+i x j i+1 x j 13+i x j 11 x 5 12+i x j 10 x 5 13+i ) for j = 2, 9 & i = 1, 3;( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x j 9 x j k ) for j = 1, 3, 4, 5, 7, 8, k = 14 & j = 2, 6, 9, k = 16;( x5i x 6 i x 7 i x 8 i x4i x 1 i x k i x 3 i ) for i = 1, 2, ..., 7, 8, k = 2, & i = 9, ..., 16, k = 9;( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x k 11 x j 16 ) , ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x k 12 x j 8 ) for (j, k) = (6, 3), (9, 6);( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x j 11 x k 10 ) , ( x j 3 x j 4 x j 13 x j 14 x j 12 x j 11 x k 13 x j 15 ) for (j, k) = (2, 7), (6, 3), (9, 6);( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x k 15 x k 16 ) for (j, k) = (2, 7), (3, 5), (6, 3), (9, 6);( x3i x 3 i+1 x 3 16−i x 3 17−i x3k1 x 3 k2 x516−i x 5 17−i ) ; for (i, k1, k2) = (3, 12, 11)(5, 15, 7), (7, 13, 15);( x j 9 x j 13 x j 10 x j 14 xk9 x j 16 x j 12 x k 14 ) for (j, k) = (2, 7), (6, 3), (9, 6);( x1i x 2 i x 3 i x 4 i x7i x 2 k x 6 i x 8 i ) for (i, k) = (1, 10), (2, 11), (3, 9), (4, 12), (5, 13), (6, 15), (7, 14), (8, 5);( x1i x 2 i x 3 i x 4 i x7i x 6 i x 3 k x 8 i ) for (i, k) = (9, 11), (10, 1), (11, 2), (12, 16), (13, 5), (14, 15), (15, 6), (16, 5);( x1i x 8 i x 2 i x 9 i x5i x 6 i x 4 i x 9 k ) for (i, k) = (9, 12), (10, 11), (11, 15), (12, 6), (13, 6), (14, 5), (15, 1), (16, 2);( x2i x 5 i x 9 i x 6 i x2k1 x 3 i x 9 k2 x6k2 ) for (i, k1, k2) = (1, 15, 10), (2, 16, 11), (3, 10, 9), (4, 1, 12), (5, 14, 13), (6, 13, 15), (7, 6, 14), (8, 14, 5);( x27 x 2 11 x 2 8 x 2 12 x29 x 2 15 x 2 10 x 5 12 ) , ( x21 x 2 7 x 2 2 x 2 8 x29 x 2 16 x 2 12 x 2 13 ) , ( x29 x 2 15 x 2 10 x 2 16 x59 x 2 12 x 5 10 x 2 11 ) , ( x211 x 2 13 x 2 12 x 2 14 x511 x 2 15 x 7 12 x 2 16 ) ,( x21 x 2 5 x 2 2 x 2 6 x23 x 2 16 x 2 9 x 2 12 ) , ( x25 x 2 6 x 2 11 x 2 12 x215 x 2 14 x 7 11 x 2 16 ) . 38 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 3.3.2. An L8- decomposition of K16�K3( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x j 5 ) , ( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) , ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 1, 2, 3;( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x j 2 x j 16 ) , ( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) for j = 1, 3;( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x 1 10 ) for j = 2, 3;( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x k2 k1 x j 1 ) for (j, k1, k2) = (1, 9, 3), (2, 9, 3), (3, 11, 3);( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x k2 k1 x j 6 ) for (j, k1, k2) = (1, 10, 1), (2, 10, 2), (3, 8, 1);( x j 3 x j 5 x j 4 x j 6 x j 2 x k2 k1 x j 9 x j 14 ) for (j, k1, k2) = (1, 11, 1), (2, 11, 2), (3, 5, 2);( x j 9 x j 15 x j 10 x j 16 x j 1 x k2 k1 x j 3 x j 14 ) for (j, k1, k2) = (1, 12, 1), (2, 15, 3), (3, 12, 3);( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x k2 k1 x j 16 ) (j, k1, k2) = (1, 4, 3), (2, 4, 1), (3, 14, 3);( x j 9 x j 13 x j 10 x j 14 x j 3 x k2 k1 x j 12 x j 7 ) for (j, k1, k2) = (1, 16, 1), (2, 13, 1), (3, 13, 1);( x13 x 1 4 x 1 13 x 1 14 x33 x 1 11 x 1 5 x 3 14 ) , ( x23 x 2 4 x 2 13 x 2 14 x33 x 3 4 x 2 5 x 3 14 ) , ( x33 x 3 4 x 3 13 x 3 14 x312 x 3 11 x 3 5 x 3 15 ) , ( x25 x 2 6 x 2 11 x 2 12 x15 x 1 6 x 2 2 x 3 12 ) ,( x21 x 2 13 x 2 2 x 2 14 x11 x 2 6 x 1 2 x 2 5 ) , ( x15 x 1 9 x 1 6 x 1 10 x35 x 1 12 x 1 8 x 1 11 ) , ( x11 x 1 7 x 1 2 x 1 8 x14 x 3 7 x 3 2 x 2 8 ) , ( x21 x 2 7 x 2 2 x 2 8 x24 x 3 7 x 3 2 x 3 8 ) ,( x31 x 3 7 x 3 2 x 3 8 x34 x 3 16 x 3 9 x 3 13 ) , ( x11 x 1 5 x 1 2 x 1 6 x31 x 1 8 x 1 12 x 1 16 ) , ( x21 x 2 5 x 2 2 x 2 6 x31 x 2 8 x 2 12 x 3 6 ) , ( x31 x 3 5 x 3 2 x 3 6 x33 x 3 8 x 3 12 x 1 6 ) ,( x13 x 1 12 x 2 12 x 2 3 x11 x 3 12 x 2 15 x 2 1 ) , ( x114 x 1 15 x 2 15 x 2 14 x14 x 3 15 x 2 5 x 2 4 ) , ( x19 x 1 11 x 2 11 x 2 9 x12 x 3 11 x 2 1 x 2 2 ) , ( x210 x 2 11 x 3 11 x 3 10 x22 x 2 4 x 3 5 x 3 8 ) ,( x17 x 1 16 x 2 16 x 2 7 x15 x 3 16 x 2 12 x 2 6 ) , ( x213 x 2 16 x 3 16 x 3 13 x28 x 2 6 x 3 6 x 3 8 ) . 3.3.3. An L8- decomposition of K16�K11( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x j 2 x j 16 ) , ( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x j 11 x j 1 ) for j = 1, 2, 4, 5, 7, 8, 10;( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) , ( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) for j = 2, 5, 6, 7, 8, 9;( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) , ( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) for j = 1, 2, 4, 5, ..., 10, 11;( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x j 16 ) , ( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) for j = 1, 2, 3, 5, ..., 11;( x3i x 7 i x 4 i x 10 i x11i x 6 i x 8 i x 9 i ) , ( x2i x 5 i x 7 i x 11 i x3i x 9 i x 8 i x 10 i ) for i = 1, 2, 5, 6, 7, 8, 10, 11, 12, 14;( x2i x 5 i x 7 i x 11 i x1i x 9 i x 8 i x 10 i ) , ( x1i x 6 i x 2 i x 7 i x3i x 5 i x 4 i x 10 i ) for i = 15, 16;( x3i x 4 i x 9 i x 8 i x5i x 6 i x 11 i x 10 i ) , ( x3i x 7 i x 4 i x 10 i x11i x 6 i x 8 i x 9 k ) for i = 15, 16;( x j 1 x j 2 x j 15 x j 16 xk1 x k 2 x j 6 x j 5 ) , ( x j 3 x j 4 x j 13 x j 14 xk3 x k 4 x k 13 x k 14 ) for (j, k) = (6, 3), (9, 3), (11, 8);( x j 5 x j 6 x j 11 x j 12 xk5 x k 6 x k 11 x k 12 ) , ( x j 7 x j 8 x j 9 x j 10 xk7 x k 8 x k 9 x k 10 ) for (j, k) = (6, 3), (9, 3), (11, 8);( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x j 5 ) for j = 2, 5, 7, 8, 10; ( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x j 9 x j 14 ) for j = 1, 2, 4, 5, ..., 10, 11; 39 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 ( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 2, 3, ..., 10, 11; ( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x j 3 x j 14 ) for j = 6, 7, 8, 9;( x j 3 x j 4 x j 13 x j 14 x j 12 x j 11 x j 5 x j 15 ) for j = 1, 2, 3, 4, 5, 7, 8, 10;( x1i x 8 i x 2 i x 9 i x4i x 6 i x 10 i x 7 i ) for i = 1, 2, 5, 6, 12, 14, 15, 16;( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) for j = 1, 2, 4, 5, ..., 11; ( x5i x 10 i x 6 i x 11 i x5k x 1 i x 9 i x 8 i ) for (i.k) = (15, 12), (16, 14);( x j 9 x j 15 x j 10 x j 16 xk9 x j 12 x j 3 x j 14 ) for (j, k) = (1, 10), (4, 1), (10, 11), (11, 3);( x j 3 x j 15 x j 4 x j 16 xk3 x j 1 x k 4 x j 2 ) for (j, k) = (10, 11), (11, 3); ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 1, 2, 3, 5, ..., 11;( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x k 15 x k 16 ) for (j, k) = (1, 11), (3, 9), (4, 5);( x j 1 x j 13 x j 2 x j 14 x j 11 x k 13 x j 10 x j 5 ) for (j, k) = (4, 1), (10, 11), (11, 3);( x j 3 x j 15 x j 4 x j 16 x k1 3 x k2 15 x k1 4 x k2 16 ) for (j, k1, k2) = (1, 10, 5), (4, 1, 11);( x1i x 6 i x 2 i x 7 i x3i x 6 k x 4 i x 10 i ) for (i, k) = (1, 12), (2, 4), (3, 12), (4, 11), (5, 15), (6, 7), (7, 13), (8, 15), (9, 11), (10, 1), (11, 2), (13, 5);( x1i x 6 i x 2 i x 7 i x3i x 6 k1 x2k2 x 10 i ) for (i, k1, k2) = (12, 16, 15), (14, 15, 16);( x3i x 4 i x 9 i x 8 i x5i x 6 i x 9 k x 10 i ) for (i, k) = (1, 12), (2, 4), (3, 12), (4, 11), (5, 15), (6, 7), (7, 13), (8, 15), (9, 11), (10, 1), (11, 2), (12, 16), (13, 5), (14, 15);( x5i x 10 i x 6 i x 11 i x8i x 1 i x 9 i x 11 k ) for (i, k) = (1, 12), (2, 4), (5, 15), (6, 7), (7, 13), (8, 15), (10, 1), (11, 2), (12, 16), (14, 15);( x5i x 10 i x 6 i x 11 i x8i x 10 k1 x9i x 11 k2 ) for (i, k1, k2) = (3, 11, 12), (4, 10, 11), (9, 1, 11), (13, 6, 5);( x1i x 8 i x 2 i x 9 i x1k x 6 i x 10 i x 7 i ) for (i, k) = (3, 11), (4, 10), (7, 5), (8, 6), (9, 1), (10, 2), (11, 1), (13, 6);( x3i x 7 i x 4 i x 10 i x3k x 6 i x 8 i x 9 i ) for (i, k) = (3, 10), (4, 1), (9, 2), (13, 7);( x2i x 5 i x 7 i x 11 i x3i x 9 i x 8 i x 11 k ) for (i, k) = (3, 11), (4, 10), (9, 1), (13, 6);( x1i x 5 i x 4 i x 11 i x2i x 6 i x 4 k x 9 i ) for (i, k) = (1, 15), (2, 16), (3, 11), (4, 10), (5, 16), (6, 15), (7, 15), (8, 16), (9, 1), (10, 12), (11, 16), (12, 4), (13, 6), (14, 8);( x35 x 3 6 x 3 11 x 3 12 x315 x 3 14 x 3 2 x 3 16 ) , ( x37 x 3 8 x 3 9 x 3 10 x36 x 3 15 x 3 11 x 3 1 ) , ( x11 x 1 13 x 1 2 x 1 14 x115 x 10 13 x 1 16 x 1 5 ) , ( x31 x 3 13 x 3 2 x 3 14 x311 x 3 6 x 3 16 x 3 5 ) ,( x33 x 3 15 x 3 4 x 3 16 x311 x 6 15 x 3 10 x 6 16 ) , ( x37 x 3 11 x 3 8 x 3 12 x39 x 3 15 x 3 5 x 3 6 ) , ( x15 x 1 9 x 1 6 x 1 10 x116 x 1 12 x 1 15 x 1 11 ) , ( x33 x 3 5 x 3 4 x 3 6 x32 x 3 11 x 3 9 x 3 16 ) ,( x31 x 3 7 x 3 2 x 3 8 x315 x 3 16 x 3 10 x 3 13 ) , ( x29 x 2 15 x 2 10 x 2 16 x21 x 3 15 x 2 3 x 3 16 ) , ( x39 x 3 15 x 3 10 x 3 16 x31 x 3 12 x 3 8 x 3 14 ) , ( x411 x 4 13 x 4 12 x 4 14 x111 x 4 15 x 2 12 x 2 14 ) ,( x43 x 4 7 x 4 4 x 4 8 x413 x 1 7 x 4 14 x 1 8 ) , ( x31 x 3 5 x 3 2 x 3 6 x33 x 3 16 x 3 12 x 3 15 ) , ( x49 x 4 13 x 4 10 x 4 14 x43 x 4 16 x 1 10 x 4 7 ) , ( x59 x 5 15 x 5 10 x 5 16 x51 x 8 15 x 5 3 x 8 16 ) . 3.3.4. An L8- decomposition of K16�K5( x j 1 x j 2 x j 15 x j 16 x51 x 5 2 x 5 15 x 5 16 ) , ( x j 3 x j 4 x j 13 x j 14 x53 x 5 4 x 5 13 x 5 14 ) for j = 1, 2, 3, 4;( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) , ( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 1, 2, 5; 40 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 ( x j 5 x j 6 x j 11 x j 12 x55 x 5 6 x 5 11 x 5 12 ) , ( x j 7 x j 8 x j 9 x j 10 x57 x 5 8 x 5 9 x 5 10 ) for j = 1, 2, 3, 4;( x j 12 x j 15 x k 15 x k 12 x j 2 x j 8 x k 14 x k 9 ) , ( x j 11 x j 16 x k 16 x k 11 x j 4 x j 6 x k 13 x k 10 ) for (j, k) = (1, 3), (2, 4);( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) , ( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) for j = 1, 2, 3, 4, 5;( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) for j = 3, 4, 5; ( x j 3 x j 7 x j 4 x j 8 x j 13 x k 7 x j 14 x j 16 ) for (j, k) = (3, 1), (4, 2);( x j 1 x j 13 x j 2 x j 14 x k2 k1 x j 6 x j 10 x j 5 ) for j = k2 = 3, 4, 5, k1 = 11&(j, k1, k2) = (1, 1, 3), (2, 1, 4);( x j 3 x j 15 x j 4 x j 16 x k2 k1 x j 1 x j 10 x j 2 ) for j = k2 = 1, 2, 5, k1 = 11&(j, k1, k2) = (3, 3, 1), (4, 3, 2);( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x k2 k1 x j 14 ) for j = k2 = 3, 4, 5, k1 = 9&(j, k1, k2) = (1, 4, 3), (2, 4, 4);( x j 9 x j 15 x j 10 x j 16 x j 1 x j k x j 3 x j 14 ) for (j, k) = (1, 5), (2, 5), (3, 7), (4, 7);( x j 11 x j 13 x j 12 x j 14 x j k x j 15 x j 4 x j 8 ) for (j, k) = (1, 1), (2, 1), (3, 3), (4, 3);( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x k 8 ) for (j, k) = (1, 3), (2, 4);( x1i x 2 i x 3 i x 4 i x1k x 2 k x 3 k x 4 k ) for (i, k) = (1, 12), (2, 4), (3, 12), (6, 7)(7, 13), (9, 11), (10, 1), (11, 2), (12, 16), (13, 5), (15, 6), (16, 5);( x1i x 2 i x 3 i x 4 i x1k1 x 2 k1 x3k2 x 4 k2 ) for (i, k1, k2) = (4, 9, 11), (5, 8, 15), (8, 16, 15), (14, 15, 7);( x51 x 5 2 x 5 15 x 5 16 x512 x 5 4 x 5 6 x 5 5 ) , ( x53 x 5 4 x 5 13 x 5 14 x512 x 5 11 x 5 5 x 5 15 ) , ( x55 x 5 6 x 5 11 x 5 12 x515 x 5 7 x 5 2 x 5 16 ) , ( x57 x 5 8 x 5 9 x 5 10 x513 x 5 15 x 5 11 x 5 1 ) ,( x35 x 3 9 x 3 6 x 3 10 x37 x 1 9 x 3 8 x 1 10 ) , ( x45 x 4 9 x 4 6 x 4 10 x47 x 2 9 x 4 8 x 2 10 ) , ( x59 x 5 15 x 5 10 x 5 16 x51 x 5 12 x 5 3 x 5 14 ) , ( x511 x 5 13 x 5 12 x 5 14 x516 x 5 15 x 5 4 x 5 8 ) ,( x53 x 5 7 x 5 4 x 5 8 x513 x 5 15 x 5 14 x 5 16 ) , ( x11 x 1 5 x 1 2 x 1 6 x13 x 3 5 x 3 2 x 3 6 ) , ( x21 x 2 5 x 2 2 x 2 6 x23 x 4 5 x 4 2 x 4 6 ) , ( x39 x 3 13 x 3 10 x 3 14 x33 x 1 13 x 3 12 x 1 14 ) ,( x49 x 4 13 x 4 10 x 4 14 x43 x 2 13 x 4 12 x 2 14 ) . 3.3.5. An L8- decomposition of K16�K13( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x j 5 ) , ( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x j 9 x j 14 ) , for j = 2, . . . , 8, 10, . . . , 13;( x j 5 x j 6 x j 11 x j 12 x j 15 x j 7 x j 2 x j 16 ) , ( x j 7 x j 8 x j 9 x j 10 x j 13 x j 15 x j 11 x j 1 ) for j = 1, . . . , 13;( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) , ( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 1, . . . , 13;( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) , ( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) for j = 2, . . . , 6, 8, 10, 12, 13;( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x j 3 x j 14 ) , ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 2, . . . 6, 8, 10, 12, 13;( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) , ( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) for j = 1, . . . , 6, 8, 9, 10, 12, 13; , ( x1i x 7 i x 2 i x 8 i x5i x 4 i x 6 i x 3 i ) , ( x4i x 5 i x 7 i x 6 i x2i x 3 i x 10 i x 1 i ) for i = 1, 2, ..., 16;( x1i x 2 i x 10 i x 9 i x4i x 12 i x 6 i x 5 i ) , ( x5i x 6 i x 13 i x 10 i x2i x 3 i x 11 i x 12 i ) , for i = 1, 2, ..., 16;( x j 7 x j 11 x j 8 x j 12 x j 14 x j 15 x j 10 x j 6 ) , ( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 16 x j 5 ) , for j = 7, 11; 41 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 ( x3i x 4 i x 8 i x 7 i x13i x 11 i x 5 i x 9 i ) , ( x2i x 3 i x 12 i x 11 i x13i x 1 i x 8 i x 10 i ) , for i = 1, 2, 5, 6, 7, 8, 11, . . . , 16;( x j 3 x j 4 x j 13 x j 14 x k1 3 x k1 4 x k2 13 x j 15 ) , ( x j 9 x j 13 x j 10 x j 14 x k1 9 x j 16 x k1 10 x k2 14 ) for (j, k1, k2) = (1, 10, 12), (9, 7, 13);( x j 3 x j 15 x j 4 x j 16 x j 11 x j 13 x j 10 x k 16 ) , ( x j 11 x j 13 x j 12 x j 14 x j 16 x k 13 x j 15 x j 8 ) for (j, k) = (7, 13), (11, 8);( x3i x 4 i x 8 i x 7 i x13i x 11 i x 5 i x 7 k ) , ( x2i x 3 i x 12 i x 11 i x13i x 1 i x 8 i x 11 k ) for (i, k) = (3, 1), (4, 12), (9, 7), (10, 2);( x j 3 x j 5 x j 4 x j 6 x j 2 x j 13 x j 9 x j 14 ) for j = 1, 9; ( x j 1 x j 2 x j 15 x j 16 x j 12 x j 4 x j 6 x k 16 ) for (j, k) = (1, 12), (9, 11);( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x j 16 ) for j = 1, . . . , 13; ( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x k 14 ) for (j, k) = (1, 11), (9, 11);( x j 1 x j 5 x j 2 x j 6 x j 15 x j 8 x j 12 x j 16 ) for j = 7, 11; ( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) for j = 2, . . . , 6, 8, 12, 13;( x j 3 x j 4 x j 13 x j 14 x j 12 x j 11 x j 5 x j 15 ) for j = 2, . . . , 8, 11, 12, 13;( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x k 14 ) for (j, k) = (7, 13), (11, 8);( x j 9 x j 15 x j 10 x j 16 x j 1 x k 15 x j 3 x j 14 ) for (j, k) = (1, 12), (7, 13), (9, 13), (11, 8);( x3i x 9 i x 4 i x 10 i x11i x 2 i x k i x 1 i ) for i = 1, 2, 5, 6, 7, 8, 11, 12, k = 12 & i = 13, 14, 15, 16 k = 13;( x3i x 9 i x 4 i x 10 i x11i x 2 i x 12 i x 10 k ) for (i, k) = (3, 12), (4, 11), (9, 3), (10, 12);( x5i x 11 i x 6 i x 12 i x13i x 7 i x 8 i x k i ) for i = 1, 2, ..., 12, k = 1 & i = 13, 14, 15, 16 k = 4;( x8i x 9 i x 12 i x 13 i x10i x 6 i x 7 i x k i ) for i = 1, 2, ..., 12, k = 4 & i = 13, 14, 15, 16 k = 1;( x1i x 11 i x 9 i x 13 i x1k x 8 i x 9 k x 7 i ) for (i, k) = (1, 15), (2, 16), (3, 12), (4, 11), (5, 16), (6, 13), (7, 14), (8, 14), (9, 3), (10, 12), (11, 5), (12, 15);( x103 x 10 4 x 10 13 x 10 14 x113 x 11 4 x 10 5 x 10 15 ) , ( x11 x 1 13 x 1 2 x 1 14 x111 x 11 13 x 1 10 x 1 5 ) , ( x91 x 9 13 x 9 2 x 9 14 x911 x 11 13 x 9 10 x 9 5 ) , ( x13 x 1 15 x 1 4 x 1 16 x111 x 11 15 x 1 10 x 11 16 ) ,( x93 x 9 15 x 9 4 x 9 16 x911 x 11 15 x 9 10 x 13 16 ) , ( x109 x 10 13 x 10 10 x 10 14 x119 x 10 16 x 11 10 x 10 7 ) . 3.3.6. An L8- decomposition of K16�K7( x j 1 x j 2 x j 15 x j 16 xk1 x k 2 x k 15 x k 16 ) , ( x j 3 x j 4 x j 13 x j 14 xk3 x k 4 x k 13 x k 14 ) for (j, k) = (1, 4), (2, 3), (3, 6), (4, 3), (5, 6), (6, 2), (7, 6);( x j 1 x j 13 x j 2 x j 14 x j 11 x j 6 x j 10 x j 5 ) , ( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 9 x j 13 ) for j = 1, 3, 6, 7;( x j 1 x j 7 x j 2 x j 8 x j 4 x j 16 x j 11 x j 13 ) for j = 4, 5; ( x j 7 x j 11 x j 8 x j 12 x j 9 x j 15 x j 10 x j 6 ) for j = 1, 3, 4, 6;( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x j 3 x j 14 ) for j = 1, 6; ( x j 5 x j 9 x j 6 x j 10 x j 7 x j 12 x j 8 x j 11 ) for j = 1, 4, 6, 7;( x j 11 x j 13 x j 12 x j 14 x j 16 x j 15 x j 4 x j 8 ) for j = 1, 6; ( x j 3 x j 7 x j 4 x j 8 x j 13 x j 15 x j 14 x j 16 ) for j = 1, 2, 4, 5, 6;( x j 3 x j 7 x j 4 x j 8 x j 10 x j 15 x j 14 x j 16 ) for j = 3, 7; ( x j 1 x j 5 x j 2 x j 6 x j 3 x j 8 x j 12 x j 16 ) for j = 1, 3, 4, 5, 6, 7;( x j 3 x j 15 x j 4 x j 16 x j 11 x j 1 x j 10 x j 2 ) for j = 1, 6, 7; ( x j 1 x j 13 x j 2 x j 14 x j 11 x k 13 x j 10 x j 5 ) for (j, k) = (2, 5), (4, 7), (5, 4);( x j 3 x j 5 x j 4 x j 6 x j 2 x j 11 x j 9 x j 14 ) for j = 1, ..., 7; ( x j 3 x j 15 x j 4 x j 16 x j 11 x k 15 x j 10 x k 16 ) (j, k) = (2, 5), (4, 7), (5, 4); 42 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 ( x j 9 x j 13 x j 10 x j 14 x j 3 x j 16 x j 12 x j 7 ) for j = 1, 3, 4, 5, 6, 7;( x j 7 x j 8 x j 9 x j 10 xk7 x k 8 x k 9 x k 10 ) for (j, k) = (1, 4), (5, 6), (6, 2), (7, 6);( x j 7 x j 8 x j 9 x j 10 x k1 7 x k2 8 x k2 9 x k2 10 ) for (j, k1, k2) = (2, 4, 3), (3, 4, 6), (4, 6, 3);( x j 9 x j 15 x j 10 x j 16 x j 1 x j 12 x k 10 x j 14 ) for (j, k) = (3, 7), (4, 7), (5, 2), (7, 2);( x j 5 x j 6 x j 11 x j 12 xk5 x k 6 x k 11 x k 12 ) for (j, k) = (1, 4), (4, 3), (5, 6), (7, 6);( x1i x 3 i x 5 i x 7 i x1k x 3 k x 5 k x 7 k ) for (i, k) = (1, 12), (2, 4), (3, 12), (4, 11), (8, 15), (9, 11), (10, 1), (12, 16), (13, 5);( x1i x 2 i x 4 i x 6 i x5i x 2 k x 4 k x 6 k ) for (i, k) = (1, 12), (2, 4), (3, 12), (4, 11), (5, 15), (8, 15), (12, 16), (14, 15);( x2i x 5 i x 4 i x 7 i x2k x 5 k x 4 k x 3 i ) for (i, k) = (1, 15), (2, 16), (4, 12), (5, 16), (6, 13), (8, 14), (9, 2);( x25 x 2 6 x 2 11 x 2 12 x27 x 3 6 x 3 11 x 3 12 ) , ( x35 x 3 6 x 3 11 x 3 12 x65 x 6 6 x 6 11 x 7 12 ) , ( x65 x 6 6 x 6 11 x 6 12 x25 x 2 6 x 2 11 x 3 12 ) , ( x33 x 3 15 x 3 4 x 3 16 x311 x 3 1 x 3 10 x 7 16 ) ,( x27 x 2 11 x 2 8 x 2 12 x216 x 7 11 x 2 6 x 7 12 ) , ( x57 x 5 11 x 5 8 x 5 12 x59 x 5 15 x 5 10 x 2 12 ) , ( x77 x 7 11 x 7 8 x 7 12 x79 x 3 11 x 7 10 x 7 6 ) , ( x25 x 2 9 x 2 6 x 2 10 x213 x 2 11 x 2 12 x 2 1 ) ,( x35 x 3 9 x 3 6 x 3 10 x316 x 3 12 x 3 8 x 3 11 ) , ( x55 x 5 9 x 5 6 x 5 10 x57 x 5 12 x 5 8 x 4 10 ) , ( x21 x 2 7 x 2 2 x 2 8 x24 x 2 6 x 2 11 x 2 10 ) , ( x29 x 2 15 x 2 10 x 2 16 x27 x 2 12 x 2 3 x 2 14 ) ,( x211 x 2 13 x 2 12 x 2 14 x216 x 2 15 x 6 12 x 5 14 ) , ( x311 x 3 13 x 3 12 x 3 14 x316 x 3 7 x 3 4 x 7 14 ) , ( x411 x 4 13 x 4 12 x 4 14 x511 x 4 15 x 7 12 x 7 14 ) , ( x511 x 5 13 x 5 12 x 5 14 x510 x 5 15 x 4 12 x 4 14 ) ,( x711 x 7 13 x 7 12 x 7 14 x716 x 2 13 x 7 4 x 7 8 ) , ( x21 x 2 5 x 2 2 x 2 6 x29 x 2 8 x 2 12 x 2 16 ) , ( x29 x 2 13 x 2 10 x 2 14 x23 x 2 16 x 2 12 x 7 14 ) , ( x15 x 3 5 x 5 5 x 7 5 x115 x 2 5 x 5 15 x 7 15 ) ,( x16 x 3 6 x 5 6 x 7 6 x17 x 3 15 x 5 12 x 7 7 ) , ( x17 x 3 7 x 5 7 x 7 7 x113 x 3 5 x 5 13 x 7 13 ) , ( x111 x 3 11 x 5 11 x 7 11 x12 x 3 2 x 2 11 x 7 2 ) , ( x114 x 3 14 x 5 14 x 7 14 x115 x 3 8 x 5 15 x 7 15 ) ,( x115 x 3 15 x 5 15 x 7 15 x16 x 3 5 x 5 6 x 7 6 ) , ( x116 x 3 16 x 5 16 x 7 16 x15 x 3 2 x 5 11 x 7 5 ) , ( x16 x 2 6 x 4 6 x 6 6 x56 x 2 15 x 4 7 x 6 7 ) , ( x19 x 2 9 x 4 9 x 6 9 x59 x 2 12 x 4 11 x 6 11 ) ,( x110 x 2 10 x 4 10 x 6 10 x510 x 2 11 x 4 1 x 6 1 ) , ( x111 x 2 11 x 4 11 x 6 11 x511 x 2 15 x 7 11 x 6 2 ) , ( x113 x 2 13 x 4 13 x 6 13 x513 x 2 8 x 4 5 x 6 5 ) , ( x115 x 2 15 x 4 15 x 6 15 x515 x 7 15 x 4 6 x 6 6 ) ,( x116 x 2 16 x 4 16 x 6 16 x516 x 7 16 x 4 11 x 6 5 ) , ( x23 x 5 3 x 4 3 x 7 3 x21 x 5 10 x 4 10 x 3 3 ) , ( x27 x 5 7 x 4 7 x 7 7 x214 x 5 6 x 4 13 x 3 7 ) , ( x17 x 2 7 x 3 7 x 6 7 x57 x 2 13 x 3 6 x 6 13 ) ,( x313 x 7 13 x 7 15 x 3 15 x33 x 7 3 x 7 11 x 3 14 ) . 3.4. L8-decomposition required for Lemma 2.7 3.4.1. An L8- decomposition of K19\K3 Let V (K19) = {x1,x2, ...,x19} and the K3 be (x13x15x17).( x1 x2 x18 x19 x9 x12 x5 x6 ) , ( x3 x4 x16 x17 x9 x1 x18 x19 ) , ( x5 x6 x14 x15 x3 x4 x16 x1 ) , ( x7 x8 x12 x13 x9 x2 x16 x14 ) ,( x9 x10 x11 x12 x4 x16 x7 x17 ) , ( x1 x16 x2 x17 x12 x3 x9 x4 ) , ( x3 x18 x4 x19 x10 x1 x2 x12 ) , ( x5 x12 x6 x13 x11 x18 x3 x16 ) ,( x7 x14 x8 x15 x2 x17 x1 x3 ) , ( x1 x10 x2 x11 x14 x13 x19 x3 ) , ( x3 x12 x4 x13 x1 x7 x11 x2 ) , ( x1 x5 x2 x6 x13 x14 x15 x11 ) ,( x7 x16 x8 x17 x6 x19 x5 x11 ) , ( x9 x18 x10 x19 x11 x6 x4 x5 ) , ( x11 x14 x12 x15 x13 x2 x10 x4 ) , ( x3 x7 x4 x8 x2 x1 x14 x9 ) ,( x5 x16 x6 x17 x7 x9 x8 x18 ) , ( x9 x14 x10 x15 x13 x3 x17 x16 ) , ( x11 x18 x13 x19 x16 x14 x8 x15 ) , ( x5 x9 x6 x10 x4 x17 x15 x8 ) , 43 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 ( x7 x18 x8 x19 x10 x15 x11 x14 ) . 3.4.2. An L8- decomposition of K15�K3( x j 7 x j 8 x j 9 x j 10 x j 15 x j 14 x j 1 x j 5 ) for j = 2, 3; ( x j 5 x j 13 x j 6 x j 15 x j 14 x j 4 x j 7 x j 2 ) for j = 1, 2, 3;( x j 3 x j 11 x j 9 x j 12 x j 8 x j 6 x j 15 x j 10 ) for j = 1, 3; ( x j 1 x j 12 x j 2 x j 13 x j 4 x j k x j 7 x j 14 ) for (j, k) = (1, 15), (2, 5), (3, 15);( x j 10 x j 13 x j 11 x j 14 x j 15 x j 8 x j k x j 1 ) for (j, k) = (1, 4), (2, 12), (3, 4);( x j 3 x j 7 x j 4 x j 9 x j 10 x j 13 x j k x j 14 ) for (j, k) = (1, 5), (2, 8), (3, 8);( x11 x 1 2 x 1 14 x 1 15 x17 x 1 8 x 1 12 x 3 15 ) , ( x21 x 2 2 x 2 14 x 2 15 x31 x 2 8 x 3 14 x 3 15 ) , ( x31 x 3 2 x 3 14 x 3 15 x37 x 3 8 x 1 14 x 3 13 ) , ( x13 x 1 4 x 1 12 x 1 13 x33 x 3 4 x 1 5 x 1 9 ) ,( x23 x 2 4 x 2 12 x 2 13 x33 x 3 4 x 2 14 x 3 13 ) , ( x33 x 3 4 x 3 12 x 3 13 x31 x 3 2 x 3 5 x 3 9 ) , ( x15 x 1 6 x 1 10 x 1 11 x13 x 2 6 x 1 2 x 1 15 ) , ( x25 x 2 6 x 2 10 x 2 11 x23 x 2 4 x 3 10 x 3 11 ) ,( x35 x 3 6 x 3 10 x 3 11 x33 x 3 4 x 3 2 x 3 15 ) , ( x17 x 1 8 x 1 9 x 1 10 x37 x 1 14 x 1 1 x 1 5 ) , ( x13 x 1 14 x 1 4 x 1 15 x12 x 2 14 x 1 10 x 1 8 ) , ( x23 x 2 14 x 2 4 x 2 15 x22 x 2 7 x 2 10 x 2 12 ) ,( x33 x 3 14 x 3 4 x 3 15 x32 x 3 7 x 3 10 x 3 8 ) , ( x15 x 1 8 x 1 6 x 1 9 x25 x 1 1 x 3 6 x 1 2 ) , ( x25 x 2 8 x 2 6 x 2 9 x24 x 2 1 x 3 6 x 3 9 ) , ( x35 x 3 8 x 3 6 x 3 9 x34 x 3 1 x 3 3 x 3 2 ) ,( x17 x 1 11 x 1 8 x 1 12 x114 x 1 1 x 3 8 x 1 6 ) , ( x27 x 2 11 x 2 8 x 2 12 x37 x 2 1 x 3 8 x 2 6 ) , ( x37 x 3 11 x 3 8 x 3 12 x39 x 3 1 x 3 10 x 3 6 ) , ( x11 x 1 5 x 1 2 x 1 6 x110 x 3 5 x 1 11 x 1 14 ) ,( x21 x 2 5 x 2 2 x 2 6 x210 x 3 5 x 2 11 x 2 14 ) , ( x31 x 3 5 x 3 2 x 3 6 x310 x 3 7 x 2 2 x 3 14 ) , ( x23 x 2 11 x 2 9 x 2 12 x28 x 2 6 x 2 15 x 3 12 ) ,( x111 x 1 12 x 3 12 x 3 11 x211 x 2 12 x 3 14 x 3 2 ) , ( x11 x 1 3 x 2 3 x 2 1 x31 x 1 6 x 2 6 x 2 7 ) , ( x12 x 1 4 x 2 4 x 2 2 x32 x 1 6 x 2 11 x 2 10 ) ,( x17 x 1 9 x 2 9 x 2 7 x15 x 3 9 x 2 2 x 2 5 ) , ( x18 x 1 10 x 2 10 x 2 8 x14 x 3 10 x 2 12 x 2 15 ) , ( x113 x 1 15 x 2 15 x 2 13 x313 x 1 7 x 2 11 x 2 9 ) . 3.5. L8-decomposition required for Lemma 2.8 3.5.1. An L8- decomposition of K13�K5( x j 5 x j 9 x j 6 x j 10 x j 3 x 3 9 x j 4 x 3 10 ) , ( x j 3 x j 12 x j 4 x j 13 x j 11 x j 2 x j 9 x j 7 ) for j = 2, 4;( x j 7 x j 8 x j 10 x j 11 x j 9 x j 2 x j 1 x j 13 ) for j = 1, 5; ( x j 1 x j 13 x j 8 x j 12 x j 7 x 3 13 x j 9 x 3 12 ) for j = 2, 5;( x j 9 x j 12 x j 10 x j 13 x j 8 x j 6 x j 4 x j 5 ) for j = 3, 4; ( x1i x 2 i x 4 i x 3 i x1k x 2 k x 4 k x 3 k ) for (i, k) = (5, 7), (8, 11);( x j 1 x j 5 x j 2 x j 6 x j k1 x k2 5 x k2 2 x k2 6 ) for (j, k1, k2) = (1, 9, 4), (2, 8, 5), (3, 9, 2);( x1i x 2 i x 5 i x 4 i x1k1 x 2 k2 x5k1 x 4 k2 ) for (i, k1, k2) = (9, 10, 11), (10, 2, 1), (13, 6, 6);( x41 x 4 5 x 4 2 x 4 6 x51 x 5 5 x 4 11 x 5 6 ) , ( x51 x 5 5 x 5 2 x 5 6 x31 x 3 5 x 3 2 x 3 6 ) , ( x13 x 1 7 x 1 4 x 1 8 x33 x 5 7 x 3 4 x 4 8 ) , ( x23 x 2 7 x 2 4 x 2 8 x21 x 4 7 x 3 4 x 5 8 ) ,( x33 x 3 7 x 3 4 x 3 8 x39 x 3 1 x 3 2 x 2 8 ) , ( x43 x 4 7 x 4 4 x 4 8 x23 x 3 7 x 3 4 x 5 8 ) , ( x53 x 5 7 x 5 4 x 5 8 x33 x 5 5 x 5 11 x 3 8 ) , ( x15 x 1 9 x 1 6 x 1 10 x55 x 3 9 x 1 12 x 3 10 ) ,( x35 x 3 9 x 3 6 x 3 10 x33 x 3 7 x 3 4 x 3 1 ) , ( x55 x 5 9 x 5 6 x 5 10 x513 x 3 9 x 1 6 x 3 10 ) , ( x11 x 1 4 x 1 5 x 1 11 x12 x 1 6 x 1 8 x 3 11 ) , ( x21 x 2 4 x 2 5 x 2 11 x22 x 4 4 x 2 8 x 5 11 ) ,( x31 x 3 4 x 3 5 x 3 11 x41 x 3 13 x 3 8 x 3 6 ) , ( x41 x 4 4 x 4 5 x 4 11 x42 x 4 3 x 4 8 x 1 11 ) , ( x51 x 5 4 x 5 5 x 5 11 x52 x 5 6 x 5 8 x 5 3 ) , ( x12 x 1 3 x 1 6 x 1 7 x111 x 1 4 x 1 5 x 1 13 ) ,( x22 x 2 3 x 2 6 x 2 7 x211 x 2 4 x 2 5 x 2 10 ) , ( x32 x 3 3 x 3 6 x 3 7 x311 x 2 3 x 3 5 x 3 10 ) , ( x42 x 4 3 x 4 6 x 4 7 x49 x 4 1 x 4 5 x 4 10 ) , ( x52 x 5 3 x 5 6 x 5 7 x511 x 5 4 x 5 5 x 3 7 ) , 44 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 ( x27 x 2 8 x 2 10 x 2 11 x29 x 2 2 x 2 3 x 2 4 ) , ( x37 x 3 8 x 3 10 x 3 11 x27 x 3 1 x 3 3 x 4 11 ) , ( x47 x 4 8 x 4 10 x 4 11 x49 x 4 1 x 4 3 x 4 4 ) , ( x11 x 1 13 x 1 8 x 1 12 x17 x 3 13 x 1 9 x 1 5 ) ,( x31 x 3 13 x 3 8 x 3 12 x33 x 3 6 x 3 2 x 3 7 ) , ( x41 x 4 13 x 4 8 x 4 12 x47 x 3 13 x 4 2 x 3 12 ) , ( x13 x 1 12 x 1 4 x 1 13 x111 x 1 2 x 5 4 x 5 13 ) , ( x33 x 3 4 x 3 12 x 3 13 x311 x 3 9 x 3 5 x 3 7 ) ,( x53 x 5 12 x 5 4 x 5 13 x51 x 5 2 x 3 4 x 5 7 ) , ( x19 x 1 12 x 1 10 x 1 13 x111 x 3 12 x 1 4 x 1 5 ) , ( x29 x 2 12 x 2 10 x 2 13 x22 x 2 6 x 2 4 x 2 5 ) , ( x59 x 5 12 x 5 10 x 5 13 x511 x 5 6 x 5 4 x 5 2 ) ,( x11 x 3 1 x 2 1 x 4 1 x13 x 3 2 x 5 1 x 4 9 ) , ( x12 x 2 2 x 4 2 x 3 2 x14 x 2 13 x 4 13 x 3 10 ) , ( x13 x 2 3 x 5 3 x 4 3 x19 x 2 9 x 5 9 x 3 3 ) , ( x14 x 2 4 x 5 4 x 4 4 x111 x 2 2 x 5 2 x 4 2 ) ,( x16 x 2 6 x 4 6 x 3 6 x111 x 2 8 x 4 8 x 3 8 ) , ( x17 x 2 7 x 5 7 x 4 7 x37 x 2 12 x 5 12 x 4 12 ) , ( x211 x 3 11 x 5 11 x 4 11 x26 x 3 13 x 5 6 x 4 6 ) , ( x112 x 2 12 x 5 12 x 4 12 x17 x 2 11 x 5 5 x 4 11 ) ,( x12 x 5 2 x 5 9 x 1 9 x113 x 4 2 x 5 4 x 1 4 ) , ( x29 x 4 9 x 4 10 x 2 10 x21 x 4 3 x 4 2 x 2 2 ) , ( x111 x 5 11 x 5 12 x 1 12 x211 x 5 8 x 5 13 x 1 13 ) , ( x212 x 4 12 x 4 13 x 2 13 x25 x 4 5 x 4 11 x 2 11 ) ,( x32 x 3 9 x 3 11 x 3 12 x313 x 3 10 x 3 4 x 3 3 ) , ( x13 x 5 3 x 5 10 x 1 10 x15 x 5 5 x 5 7 x 1 7 ) , ( x11 x 5 1 x 5 8 x 1 8 x21 x 5 9 x 5 6 x 1 6 ) . 3.6. L8-decomposition required for Lemma 2.9 3.6.1. An L8- decomposition of K11�K7( x j 1 x j 5 x j 2 x j 6 x j 9 x j 7 x j 4 x j 8 ) for j = 2, 5; ( x j 1 x j 4 x j 5 x j 11 x j 3 x j 9 x j 8 x k 11 ) for (j, k) = (2, 3), (3, 4);( x j 5 x j 9 x j 6 x j 10 x j 3 x j 11 x j 4 x j 1 ) for j = 5, 6; ( x j 7 x j 8 x j 10 x j 11 x j 9 x j 1 x j 2 x j 4 ) for j = 1, 2, 5;( x j 2 x j 3 x j 6 x j 7 x j 1 x j 11 x j 5 x k 7 ) for (j, k) = (5, 2), (6, 2);( x j 7 x j 8 x j 10 x j 11 x j 9 x j 1 x k 10 x j 4 ) for (j, k) = (3, 6), (6, 7);( x1i x 3 i x 5 i x 7 i x4i x 2 i x 6 i x 7 k ) for (i, k) = (5, 9), (6, 8), (7, 1), (8, 9);( x1i x 3 i x 5 i x 7 i x1k1 x 2 i x 6 i x 7 k2 ) for (i, k1, k2) = (1, 9, 9), (4, 3, 3), (9, 8, 2);( x1i x 2 i x 4 i x 6 i x5i x 2 k x 3 i x 7 i ) for (i, k) = (4, 3), (8, 9), (9, 2);( x2i x 5 i x 4 i x 7 i x6i x 5 k x 4 k x 3 i ) for (i, k) = (3, 10), (8, 9);( x11 x 1 5 x 1 2 x 1 6 x110 x 1 3 x 1 9 x 1 8 ) , ( x31 x 3 5 x 3 2 x 3 6 x39 x 3 7 x 3 10 x 3 11 ) , ( x41 x 4 5 x 4 2 x 4 6 x11 x 4 10 x 1 2 x 4 8 ) , ( x61 x 6 5 x 6 2 x 6 6 x31 x 3 5 x 2 2 x 3 6 ) ,( x71 x 7 5 x 7 2 x 7 6 x61 x 6 5 x 6 2 x 7 11 ) , ( x13 x 1 7 x 1 4 x 1 8 x111 x 1 1 x 1 10 x 1 2 ) , ( x23 x 2 7 x 2 4 x 2 8 x29 x 2 1 x 2 10 x 2 2 ) , ( x33 x 3 7 x 3 4 x 3 8 x35 x 3 1 x 3 10 x 3 2 ) ,( x43 x 4 7 x 4 4 x 4 8 x33 x 4 1 x 4 9 x 4 2 ) , ( x53 x 5 7 x 5 4 x 5 8 x59 x 5 1 x 5 10 x 5 5 ) , ( x63 x 6 7 x 6 4 x 6 8 x69 x 3 7 x 6 10 x 6 5 ) , ( x73 x 7 7 x 7 4 x 7 8 x75 x 6 7 x 7 6 x 7 2 ) ,( x15 x 1 9 x 1 6 x 1 10 x18 x 1 11 x 1 4 x 4 10 ) , ( x25 x 2 9 x 2 6 x 2 10 x23 x 2 11 x 2 4 x 3 10 ) , ( x35 x 3 9 x 3 6 x 3 10 x75 x 6 9 x 3 4 x 3 1 ) , ( x77 x 7 8 x 7 10 x 7 11 x79 x 7 1 x 7 2 x 7 3 ) ,( x11 x 1 4 x 1 5 x 1 11 x13 x 1 9 x 1 7 x 1 8 ) , ( x41 x 4 4 x 4 5 x 4 11 x43 x 4 10 x 4 8 x 4 9 ) , ( x51 x 5 4 x 5 5 x 5 11 x53 x 5 9 x 2 5 x 5 6 ) , ( x61 x 6 4 x 6 5 x 6 11 x13 x 6 9 x 2 5 x 2 11 ) ,( x71 x 7 4 x 7 5 x 7 11 x73 x 7 2 x 7 8 x 6 11 ) , ( x12 x 1 3 x 1 6 x 1 7 x111 x 1 9 x 1 5 x 1 10 ) , ( x22 x 2 3 x 2 6 x 2 7 x21 x 3 3 x 2 5 x 2 10 ) , ( x32 x 3 3 x 3 6 x 3 7 x31 x 3 9 x 3 5 x 3 10 ) ,( x42 x 4 3 x 4 6 x 4 7 x41 x 4 11 x 4 9 x 4 10 ) , ( x72 x 7 3 x 7 6 x 7 7 x32 x 7 9 x 7 5 x 7 10 ) , ( x12 x 3 2 x 5 2 x 7 2 x14 x 4 2 x 5 8 x 7 11 ) , ( x13 x 3 3 x 5 3 x 7 3 x110 x 3 4 x 6 3 x 7 10 ) ,( x110 x 3 10 x 5 10 x 7 10 x19 x 3 3 x 5 7 x 7 9 ) , ( x111 x 3 11 x 5 11 x 7 11 x411 x 3 9 x 6 11 x 7 8 ) , ( x11 x 2 1 x 4 1 x 6 1 x12 x 2 10 x 3 1 x 6 9 ) , ( x12 x 2 2 x 4 2 x 6 2 x52 x 2 11 x 4 9 x 6 8 ) ,( x13 x 2 3 x 4 3 x 6 3 x53 x 2 10 x 4 9 x 7 3 ) , ( x15 x 2 5 x 4 5 x 6 5 x55 x 7 5 x 3 5 x 6 7 ) , ( x16 x 2 6 x 4 6 x 6 6 x111 x 2 11 x 3 6 x 6 8 ) , ( x17 x 2 7 x 4 7 x 6 7 x57 x 7 7 x 3 7 x 6 1 ) ,( x110 x 2 10 x 4 10 x 6 10 x510 x 2 9 x 3 10 x 6 7 ) , ( x111 x 2 11 x 4 11 x 6 11 x511 x 2 8 x 4 10 x 6 6 ) , ( x21 x 5 1 x 4 1 x 7 1 x61 x 1 1 x 4 9 x 3 1 ) , ( x22 x 5 2 x 4 2 x 7 2 x32 x 5 11 x 4 11 x 7 1 ) , 45 K. Sowndhariya, A. Muthusamy / J. Algebra Comb. Discrete Appl. 9(1) (2022) 29–46 ( x24 x 5 4 x 4 4 x 7 4 x64 x 5 3 x 1 4 x 3 4 ) , ( x62 x 6 4 x 6 3 x 6 10 x52 x 3 4 x 3 3 x 5 10 ) , ( x26 x 5 6 x 4 6 x 7 6 x66 x 1 6 x 4 11 x 3 6 ) , ( x29 x 5 9 x 4 9 x 7 9 x69 x 5 2 x 1 9 x 7 11 ) ,( x210 x 5 10 x 4 10 x 7 10 x610 x 5 9 x 4 1 x 3 10 ) , ( x211 x 5 11 x 4 11 x 7 11 x23 x 5 8 x 4 8 x 3 11 ) , ( x32 x 3 9 x 3 8 x 3 11 x34 x 3 10 x 3 6 x 3 3 ) , ( x62 x 6 9 x 6 8 x 6 11 x32 x 6 10 x 3 8 x 3 11 ) ( x43 x 4 4 x 4 6 x 4 5 x13 x 1 4 x 4 10 x 4 9 ) , ( x47 x 4 8 x 4 10 x 4 9 x411 x 4 1 x 4 2 x 4 4 ) , ( x74 x 7 9 x 7 6 x 7 10 x711 x 3 9 x 6 6 x 7 1 ) , ( x45 x 4 7 x 7 7 x 7 5 x55 x 5 7 x 3 7 x 7 10 ) . 3.7. L8-decomposition required for Lemma 2.10 3.7.1. An L8- decomposition of K9�K9( x j 1 x j 2 x j 3 x j 4 x j 7 x j 5 x j 6 x j 8 ) for j = 4, 6, 7, 8; ( x j 5 x j 6 x j 7 x j 8 x j 4 x j 1 x j 2 x j 3 ) for j = 1, 2, 4, 5, 6, 7, 8;( x1i x 2 i x 3 i x 4 i x7i x 5 i x 6 i x 8 i ) for i = 2, 4, 6; ( x j 1 x j 8 x j 2 x j 9 x j 5 x j 6 x j 4 x j 7 ) for j = 1, 2, 3, 4, 7, 8, 9;( x1i x 2 i x 3 i x 4 i x7i x 6 i x 3 k x 8 i ) for (i, k) = (1, 7), (7, 2); ( x1i x 2 i x 3 i x 4 i x7i x 2 k x 6 i x 8 i ) for (i, k) = (3, 5), (8, 4);( x1i x 2 i x 3 i x 4 i x1k x 5 i x 6 i x 4 k ) for (i, k) = (5, 3), (9, 6); ( x5i x 6 i x 7 i x 8 i x4i x 1 i x 2 i x 3 i ) for i = 1, 3, 4, 6, 7, 8;( x1i x 8 i x 2 i x 9 i x5i x 6 i x 4 i x 7 i ) for i = 1, 3, . . . , 9; ( x3i x 7 i x 4 i x 9 i x1i x 5 i x 6 i x 8 i ) for i = 1, 3, 4, 5, 7, 8, 9;( x j 3 x j 7 x j 4 x j 9 x j 1 x j 5 x j 6 x j 8 ) for j = 1, 2, 3, 4, 6, 7, 8, 9;( x11 x 1 2 x 1 3 x 1 4 x17 x 5 2 x 1 6 x 1 8 ) , ( x21 x 2 2 x 2 3 x 2 4 x27 x 2 5 x 2 6 x 6 4 ) , ( x31 x 3 2 x 3 3 x 3 4 x61 x 8 2 x 3 6 x 3 8 ) , ( x51 x 5 2 x 5 3 x 5 4 x31 x 7 2 x 5 6 x 3 4 ) ,( x91 x 9 2 x 9 3 x 9 4 x61 x 9 5 x 9 6 x 6 4 ) , ( x35 x 3 6 x 3 7 x 3 8 x34 x 3 1 x 6 7 x 5 8 ) , ( x95 x 9 6 x 9 7 x 9 8 x55 x 9 9 x 9 2 x 9 3 ) , ( x51 x 5 8 x 5 2 x 5 9 x55 x 5 6 x 3 2 x 5 7 ) ,( x61 x 6 8 x 6 2 x 6 9 x65 x 2 8 x 6 4 x 6 7 ) , ( x53 x 5 7 x 5 4 x 5 9 x51 x 2 7 x 5 6 x 5 8 ) , ( x52 x 6 2 x 7 2 x 8 2 x42 x 1 2 x 2 2 x 8 6 ) , ( x55 x 6 5 x 7 5 x 8 5 x57 x 1 5 x 2 5 x 3 5 ) ,( x59 x 6 9 x 7 9 x 8 9 x55 x 9 9 x 2 9 x 3 9 ) , ( x12 x 8 2 x 2 2 x 9 2 x16 x 6 2 x 4 2 x 7 2 ) , ( x32 x 7 2 x 4 2 x 9 2 x12 x 7 6 x 6 2 x 8 2 ) , ( x36 x 7 6 x 4 6 x 9 6 x16 x 5 6 x 4 2 x 8 6 ) ,( x26 x 6 6 x 6 9 x 2 9 x22 x 6 8 x 6 5 x 2 5 ) , ( x36 x 5 6 x 5 9 x 3 9 x32 x 5 2 x 9 9 x 3 5 ) , ( x45 x 8 5 x 8 9 x 4 9 x55 x 8 3 x 8 6 x 5 9 ) , ( x15 x 7 5 x 7 9 x 1 9 x12 x 7 3 x 7 6 x 6 9 ) ,( x63 x 9 3 x 9 5 x 6 5 x23 x 5 3 x 9 9 x 2 5 ) , ( x33 x 5 3 x 5 5 x 3 5 x38 x 2 3 x 5 2 x 3 2 ) , ( x62 x 9 2 x 9 6 x 6 6 x22 x 5 2 x 5 6 x 4 6 ) , ( x51 x 9 1 x 9 7 x 5 7 x21 x 9 6 x 6 7 x 3 7 ) ,( x54 x 9 4 x 9 8 x 5 8 x52 x 9 5 x 6 8 x 2 8 ) . 46 Introduction Decomposition of Km Kn into sunlet graph of order 8 References Appendix