- Email: [email protected]
Metamorphosis problems for block designs
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 40 (2013) 199–204 www.elsevier.com/locate/endm
Metamorphosis problems for block designs Selda K¨ u¸cu ¨k¸cif¸ci 1 Department of Mathematics Ko¸c University Istanbul, Turkey
Abstract In this paper metamorphosis problems for block designs will be presented and recent developments will be surveyed. Keywords: Metamorphosis of G-designs, full metamorphosis, simultaneous metamorphosis.
1
Introduction
A λ-fold G-design of order n is a pair (X, B), where X is a set of n vertices and B is a collection of edge disjoint copies of the simple graph G, called blocks, which partitions the edge set of λKn (the undirected complete graph with n vertices) with vertex set X. When G = Kk the λ-fold G-design is called a λ-fold block design of order n with block size k. Let (X, B) be a G-design and H be a subgraph of G. For each block b ∈ B, partition b into copies of H and G \ H and place the copy of H in B(H) and the edges belonging to the copy of G \ H in D(G \ H). Now if the edges belonging to D(G \ H) can be arranged into a collection D(H) of copies of 1
Email: [email protected]
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.036
200
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
H, then (X, B(H) ∪ D(H)) is a λ-fold H-design of order n and is called a metamorphosis of the λ-fold G-design (X, B) into a λ-fold H-design. The first papers, in which the term “metamorphosis” was introduced for graph designs, were published in the beginning of 21st century for the case (G, H) = (K4 , C4 ) by Lindner and Street [13] (where C4 is a 4-cycle) and for the case (G, H) = (K4 , K3 ) by Lindner and Rosa [14]. But the metamorphosis problem was first considered from a different perspective by Bryant earlier in 1996 in [3] while solving a problem on nested Steiner triple systems. Then various researchers; Billington, Dancer, K¨ u¸cu ¨k¸cif¸ci, Lindner and Rosa contributed to the problem and gave constructions for G = K4 and all possible subgraphs H of G in a series of papers. Afterwards, variations of the metamorphosis problems such as full metamorphosis, simultaneous metamorphosis and metamorphosis of maximum packings have been studied. In this work, these problems will be presented and recent developments will be explained, when G is a connected subgraph of K4 . Before presenting the results we need to give few more definitions and notations. The metamorphosis of λ-fold G-design into a λ-fold H-design will be denoted by (G > H) − Mλ (n), where n is the order of the G-design. The set of all integers n such that there exists a (G > H) − Mλ (n) will be denoted by Meta(G > H, λ). An m-cycle (m ≥ 3) on the vertices v1 , v2 , ..., vm with the edge set {{vi , vi+1 }| 1 ≤ i ≤ m−1}∪{v1 , vm } will be denoted by (v1 , v2 , ..., vm ) or (v1 , vm , vm−1 , ..., v2 ) or any cyclic shift of these. A K1,t star is the complete bipartite graph with the vertex set {c} ∪ {v1 , v2 , ..., vt } and the edge set {{c, v1 }, {c, v2 },..., {c, vt }}. A k-path, Pk is a simple graph with k vertices, v1 , v2 , ..., vk and edges {vi , vi+1 }, 1 ≤ i ≤ k − 1 with vi 6= vj , if i 6= j. A kite Te is a simple graph on 4 vertices consisting of a triangle with a pendant edge.
2
Metamorphosis problems for block designs
In this section, we will summarize the results on the metamorphosis of a λ-fold K4 -design into a λ-fold H-design, that is, the existence of a (K4 > H)−Mλ (n) for all possible subgraphs H of K4 . First, consider the following example to illustrate the definition of a (K4 > H) − Mλ (n). Example 2.1 (K4 > K3 ) − Mλ (13) (Metamorphosis of a K4 -design of order
201
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
13 into a triple system.) Let (Z13 , B) be the K4 -design of order 13 with the base block {0, 1, 5, 11} (mod 13). Put the star isomorphic to K1,3 centered at 1 + i, for i ∈ Z13 from each cyclic shift of {0, 1, 5, 11} in D(G \ H) and rearrange the edges in D(G \ H) to obtain D(H) = {(i, 1 + i, 4 + i) | 0 ≤ i ≤ 12} (mod 13). Then (Z13 , B(H) ∪ D(H)) forms a K3 -design of order 13. In Table 1 we list all the connected subgraphs H of K4 , where the existence of a (K4 > H) − Mλ (n) has been proven. H
|E(H)|
References
K4 − e
5
[7,15]
Te
4
[6]
C4
4
[13]
K3
3
[14]
K1,3
3
[2]
P4
3
trivial
P3
2
trivial
Table 1: Meta(K4 > H, λ), where H
(G, H)
K4 − e
C4
Te
K3
P4
K1,3
P3
2P2
P2
K4 − e
−
[16,18]
[4]
[21]
[22]
[22]
[22]
[22]
*
C4
×
−
×
×
[21]
×
*
*
*
Te
×
×
−
[17]
[19]
[21]
*
[21]
*
K3
×
×
×
−
×
×
[21]
×
*
P4
×
×
×
×
−
×
[21]
[21]
*
K1,3
×
×
×
×
×
−
[21]
×
*
P3
×
×
×
×
×
×
−
×
*
2P2
×
×
×
×
×
×
×
−
*
Table 2: Meta(G > H, λ) where G is a proper subgraph of K4
is a connected subgraph of K4
These papers listed in Table 1 complete the problem for G = K4 and all possible connected subgraphs H of G. The main ingredients in the constructions used to complete the problem are skew Room squares. The constructions include also pairwise balanced designs, commutative quasigroups, self-orthogonal quasigroups and transversal designs. Next, the problem was considered when G is a possible non-trivial subgraph of K4 . Table 2 summarizes those results. In Table 2 “×” denotes that H is not a subgraph of G and “∗” means that the construction of a (G > H) − Mλ (n) is trivial. Packings of metamorphoses of a λ-fold block design into a λ-fold H-design for H = C4 and H = T3 have also been studied by K¨ u¸cu ¨k¸cif¸ci, et al. in [8] and by K¨ u¸cu ¨k¸cif¸ci in [9], respectively.
202
3
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
Variations of metamorphosis problems for block designs
In this section we will present very recent works on the generalization of the metamorphoses problems explained in the previous section. Note that when we consider a (K4 > K3 ) − Mλ (n), K4 \ K3 can be 4 different K1,3 . This lead us to the following definition: Let (X, B) be a λ-fold K4 -design and label the elements of each block b with b1 , b2 , b3 , and b4 (in any manner). For each i = 1, 2, 3, 4 define a set of triangles Hi and a set of stars G\Hi as follows: for each block b = [b1 , b2 , b3 , b4 ] belonging to B, partition the edges of b into a triangle and a star centered at bi , and place the triangle in B(Hi ) and the star in D(G \ Hi ). Now if the stars in D(G \ Hi ) can be arranged into a collection of triples D(Hi ), then (X, B(Hi ) ∪ D(Hi )) is a (K4 > K3 ) − Mλ (n). We will refer to Mi = (X, B(Hi ) ∪ D(Hi )) as the ith metamorphosis of (X, B). Observe that the center of the star corresponding to each block b is different in each metamorphosis Mi . The full metamorphosis of a K4 -design (X, B) into a λ-fold triple system is a set of four metamorphoses {M1 , M2 , M3 , M4 } and will be denoted by full−(K4 > K3 ) − Mλ (n). The problem of determining the existence of a full−(K4 > K3 )−Mλ (n) for all possible values of n and λ has been completely solved by K¨ u¸cu ¨k¸cif¸ci et al. in [11]. Similarly, a full−(K4 > C4 ) − Mλ (n) and a full−(K4 > T3 ) − Mλ (n) have been defined and their existence for all possible values of n and λ have been completely solved by K¨ u¸cu ¨k¸cif¸ci et al. in [10] and [12]. The constructions in [10,11,12] use group divisible designs and pairwise balanced designs. In 2003, Adams, Billington and Mahmoodian introduced a new perspective to the metamorphosis problems for graph designs. Let Hi , 1 ≤ i ≤ r be subgraphs of the graph G. If there are metamorphoses Meta(G > Hi , λ)s with the same set of G-blocks, then we call them simultaneous metamorphoses of λ-fold G-designs. In [1] the simultaneous metamorphosis of K4 -designs into Hi -designs (i = 1, 2) for H1 = K1,3 and H2 = C3 , when λ = 1 have been constructed by Adams et al. Very recently, G. Ragusa [20] gave a complete answer to the existence of a λ-fold kite design having a complete simultaneous metamorphosis into Hi -designs (i = 1, 2, 3) when H1 = K1,3 , H2 = C3 and H3 = P4 . These are the two papers published up to date on the simultaneous metamorphosis problems for block designs.
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
203
References [1] P. Adams, E. J. Billington and E. S. Mahmoodian, The simultaneous metamorphosis of small-wheel systems, J. Combin. Math. Combin. Comput. 44 (2003), 209–223. [2] E. J. Billington and K. A. Dancer, The metamorphosis of designs with block size four: a survey and the final case, Congr. Numer. 164 (2003), 129–151. [3] D. E. Bryant, A special class of nested Steiner triple systems, Discrete Math. 152 (1996), 315–320. [4] Y. Chang, G. Lo Faro and A. Tripodi, The spectrum of Meta(K4 −e > K3 +e, λ) with any λ, Util. Math. 72 (2007), 3–22. [5] Y. Chang, T. Feng, G. Lo Faro and A. Tripodi, The existence spectrum of (3) (3) Meta(K4 > K4 − e), Sci. China Math. 53(11) (2010), 2865–2876. [6] S. K¨ u¸cu ¨k¸cif¸ci and C. C. Lindner, The metamorphosis of λ-fold block designs with block size four into λ-fold kite systems, J. Combin. Math. Combin. Comput. 40 (2002), 241–252. [7] S. K¨ u¸cu ¨k¸cif¸ci and C. C. Lindner, The metamorphosis of λ-fold block designs with block size four into K4 \ e designs, λ ≥ 2, Util. Math. 63 (2003), 239–254. [8] S. K¨ u¸cu ¨k¸cif¸ci, C. C. Lindner, and A. Rosa The metamorphosis of λ-fold block designs with block size four into a maximum packing of λKn with 4-cycles, Discrete Math. 278(1–3) (2004), 175–193. [9] S. K¨ u¸cu ¨k¸cif¸ci, The metamorphosis of λ-fold block designs with block size four into maximum packings of λKn with kites, Util. Math. 68 (2005), 165–195. [10] S. K¨ u¸cu ¨k¸cif¸ci, C. C. Lindner and E. S¸. Yazıcı, The full metamorphosis of λ-fold block designs with block size four into a λ-fold 4-cycle systems, Ars Combin. 104 (2012), 81–96. [11] S. K¨ u¸cu ¨k¸cif¸ci, C. C. Lindner and E. S¸. Yazıcı, The full metamorphosis of λ-fold block designs with block size four into a λ-fold triple systems, Ars Combin. 106 (2012), 337–351. [12] S. K¨ u¸cu ¨k¸cif¸ci, B. Smith and E. S¸. Yazıcı, The full metamorphosis of λ-fold block designs with block size four into a λ-fold kite systems, Util. Math., to appear. [13] C. C. Lindner and A. Street, The metamorphosis of λ-fold block designs with block size four into λ-fold 4-cycle systems, Bulletin of the ICA 28 (2000), 7-18.
204
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
[14] C. C. Lindner and A. Rosa, The metamorphosis of λ-fold block designs with block size four into λ-fold triple systems, J. Statist. Plann. Inference 106(1–2) (2002), 69–76. [15] C. C. Lindner and A. Rosa, The metamorphosis of block designs with block size four into (K4 \ e)-designs, Util. Math. 61 (2002), 33–46. [16] C. C. Lindner and A. Tripodi, The metamorphosis of K4 \ e-designs into maximum packings of Kn with 4-cycles, Ars Combin. 75 (2005), 333–349. [17] C. C. Lindner, G. Lo Faro and A. Tripodi, The metamorphosis of λ-fold kite systems into maximum packings of λKn with triangles, J. Combin. Math. Combin. Comput. 56 (2006), 171–189. [18] C. C. Lindner and A. Tripodi, The metamorphosis of λ-fold K4 − e designs into maximum packings of λKn with 4-cycles, J. Statist. Plann. Inference 138(11) (2008), 3316–3325. [19] G. Lo Faro and A. Tripodi, The spectrum of Meta(K3 + e > P4 , λ) and Meta(K3 + e > H4 , λ) with any λ, Discrete Math. 309(2) (2009), 354–362. [20] G. Ragusa, Complete simultaneous metamorphosis of λ-fold kite systems, J. Combin. Math. Combin. Comput. 73 (2010), 159–180. [21] L. Yuan, Y. Zhang, Y. Hou and Q. Kang, The spectrum of M eta(H > T, λ) for all subgraphs H and T of K4 (I), to appear. [22] Y. Zhang, Y. Hou and Q. Kang, The spectrum of M eta(H > T, λ) for all subgraphs H and T of K4 (II), Bulletin of the ICA 59 (2010), 46-70.
Electronic Notes in Discrete Mathematics 40 (2013) 199–204 www.elsevier.com/locate/endm
Metamorphosis problems for block designs Selda K¨ u¸cu ¨k¸cif¸ci 1 Department of Mathematics Ko¸c University Istanbul, Turkey
Abstract In this paper metamorphosis problems for block designs will be presented and recent developments will be surveyed. Keywords: Metamorphosis of G-designs, full metamorphosis, simultaneous metamorphosis.
1
Introduction
A λ-fold G-design of order n is a pair (X, B), where X is a set of n vertices and B is a collection of edge disjoint copies of the simple graph G, called blocks, which partitions the edge set of λKn (the undirected complete graph with n vertices) with vertex set X. When G = Kk the λ-fold G-design is called a λ-fold block design of order n with block size k. Let (X, B) be a G-design and H be a subgraph of G. For each block b ∈ B, partition b into copies of H and G \ H and place the copy of H in B(H) and the edges belonging to the copy of G \ H in D(G \ H). Now if the edges belonging to D(G \ H) can be arranged into a collection D(H) of copies of 1
Email: [email protected]
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.036
200
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
H, then (X, B(H) ∪ D(H)) is a λ-fold H-design of order n and is called a metamorphosis of the λ-fold G-design (X, B) into a λ-fold H-design. The first papers, in which the term “metamorphosis” was introduced for graph designs, were published in the beginning of 21st century for the case (G, H) = (K4 , C4 ) by Lindner and Street [13] (where C4 is a 4-cycle) and for the case (G, H) = (K4 , K3 ) by Lindner and Rosa [14]. But the metamorphosis problem was first considered from a different perspective by Bryant earlier in 1996 in [3] while solving a problem on nested Steiner triple systems. Then various researchers; Billington, Dancer, K¨ u¸cu ¨k¸cif¸ci, Lindner and Rosa contributed to the problem and gave constructions for G = K4 and all possible subgraphs H of G in a series of papers. Afterwards, variations of the metamorphosis problems such as full metamorphosis, simultaneous metamorphosis and metamorphosis of maximum packings have been studied. In this work, these problems will be presented and recent developments will be explained, when G is a connected subgraph of K4 . Before presenting the results we need to give few more definitions and notations. The metamorphosis of λ-fold G-design into a λ-fold H-design will be denoted by (G > H) − Mλ (n), where n is the order of the G-design. The set of all integers n such that there exists a (G > H) − Mλ (n) will be denoted by Meta(G > H, λ). An m-cycle (m ≥ 3) on the vertices v1 , v2 , ..., vm with the edge set {{vi , vi+1 }| 1 ≤ i ≤ m−1}∪{v1 , vm } will be denoted by (v1 , v2 , ..., vm ) or (v1 , vm , vm−1 , ..., v2 ) or any cyclic shift of these. A K1,t star is the complete bipartite graph with the vertex set {c} ∪ {v1 , v2 , ..., vt } and the edge set {{c, v1 }, {c, v2 },..., {c, vt }}. A k-path, Pk is a simple graph with k vertices, v1 , v2 , ..., vk and edges {vi , vi+1 }, 1 ≤ i ≤ k − 1 with vi 6= vj , if i 6= j. A kite Te is a simple graph on 4 vertices consisting of a triangle with a pendant edge.
2
Metamorphosis problems for block designs
In this section, we will summarize the results on the metamorphosis of a λ-fold K4 -design into a λ-fold H-design, that is, the existence of a (K4 > H)−Mλ (n) for all possible subgraphs H of K4 . First, consider the following example to illustrate the definition of a (K4 > H) − Mλ (n). Example 2.1 (K4 > K3 ) − Mλ (13) (Metamorphosis of a K4 -design of order
201
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
13 into a triple system.) Let (Z13 , B) be the K4 -design of order 13 with the base block {0, 1, 5, 11} (mod 13). Put the star isomorphic to K1,3 centered at 1 + i, for i ∈ Z13 from each cyclic shift of {0, 1, 5, 11} in D(G \ H) and rearrange the edges in D(G \ H) to obtain D(H) = {(i, 1 + i, 4 + i) | 0 ≤ i ≤ 12} (mod 13). Then (Z13 , B(H) ∪ D(H)) forms a K3 -design of order 13. In Table 1 we list all the connected subgraphs H of K4 , where the existence of a (K4 > H) − Mλ (n) has been proven. H
|E(H)|
References
K4 − e
5
[7,15]
Te
4
[6]
C4
4
[13]
K3
3
[14]
K1,3
3
[2]
P4
3
trivial
P3
2
trivial
Table 1: Meta(K4 > H, λ), where H
(G, H)
K4 − e
C4
Te
K3
P4
K1,3
P3
2P2
P2
K4 − e
−
[16,18]
[4]
[21]
[22]
[22]
[22]
[22]
*
C4
×
−
×
×
[21]
×
*
*
*
Te
×
×
−
[17]
[19]
[21]
*
[21]
*
K3
×
×
×
−
×
×
[21]
×
*
P4
×
×
×
×
−
×
[21]
[21]
*
K1,3
×
×
×
×
×
−
[21]
×
*
P3
×
×
×
×
×
×
−
×
*
2P2
×
×
×
×
×
×
×
−
*
Table 2: Meta(G > H, λ) where G is a proper subgraph of K4
is a connected subgraph of K4
These papers listed in Table 1 complete the problem for G = K4 and all possible connected subgraphs H of G. The main ingredients in the constructions used to complete the problem are skew Room squares. The constructions include also pairwise balanced designs, commutative quasigroups, self-orthogonal quasigroups and transversal designs. Next, the problem was considered when G is a possible non-trivial subgraph of K4 . Table 2 summarizes those results. In Table 2 “×” denotes that H is not a subgraph of G and “∗” means that the construction of a (G > H) − Mλ (n) is trivial. Packings of metamorphoses of a λ-fold block design into a λ-fold H-design for H = C4 and H = T3 have also been studied by K¨ u¸cu ¨k¸cif¸ci, et al. in [8] and by K¨ u¸cu ¨k¸cif¸ci in [9], respectively.
202
3
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
Variations of metamorphosis problems for block designs
In this section we will present very recent works on the generalization of the metamorphoses problems explained in the previous section. Note that when we consider a (K4 > K3 ) − Mλ (n), K4 \ K3 can be 4 different K1,3 . This lead us to the following definition: Let (X, B) be a λ-fold K4 -design and label the elements of each block b with b1 , b2 , b3 , and b4 (in any manner). For each i = 1, 2, 3, 4 define a set of triangles Hi and a set of stars G\Hi as follows: for each block b = [b1 , b2 , b3 , b4 ] belonging to B, partition the edges of b into a triangle and a star centered at bi , and place the triangle in B(Hi ) and the star in D(G \ Hi ). Now if the stars in D(G \ Hi ) can be arranged into a collection of triples D(Hi ), then (X, B(Hi ) ∪ D(Hi )) is a (K4 > K3 ) − Mλ (n). We will refer to Mi = (X, B(Hi ) ∪ D(Hi )) as the ith metamorphosis of (X, B). Observe that the center of the star corresponding to each block b is different in each metamorphosis Mi . The full metamorphosis of a K4 -design (X, B) into a λ-fold triple system is a set of four metamorphoses {M1 , M2 , M3 , M4 } and will be denoted by full−(K4 > K3 ) − Mλ (n). The problem of determining the existence of a full−(K4 > K3 )−Mλ (n) for all possible values of n and λ has been completely solved by K¨ u¸cu ¨k¸cif¸ci et al. in [11]. Similarly, a full−(K4 > C4 ) − Mλ (n) and a full−(K4 > T3 ) − Mλ (n) have been defined and their existence for all possible values of n and λ have been completely solved by K¨ u¸cu ¨k¸cif¸ci et al. in [10] and [12]. The constructions in [10,11,12] use group divisible designs and pairwise balanced designs. In 2003, Adams, Billington and Mahmoodian introduced a new perspective to the metamorphosis problems for graph designs. Let Hi , 1 ≤ i ≤ r be subgraphs of the graph G. If there are metamorphoses Meta(G > Hi , λ)s with the same set of G-blocks, then we call them simultaneous metamorphoses of λ-fold G-designs. In [1] the simultaneous metamorphosis of K4 -designs into Hi -designs (i = 1, 2) for H1 = K1,3 and H2 = C3 , when λ = 1 have been constructed by Adams et al. Very recently, G. Ragusa [20] gave a complete answer to the existence of a λ-fold kite design having a complete simultaneous metamorphosis into Hi -designs (i = 1, 2, 3) when H1 = K1,3 , H2 = C3 and H3 = P4 . These are the two papers published up to date on the simultaneous metamorphosis problems for block designs.
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
203
References [1] P. Adams, E. J. Billington and E. S. Mahmoodian, The simultaneous metamorphosis of small-wheel systems, J. Combin. Math. Combin. Comput. 44 (2003), 209–223. [2] E. J. Billington and K. A. Dancer, The metamorphosis of designs with block size four: a survey and the final case, Congr. Numer. 164 (2003), 129–151. [3] D. E. Bryant, A special class of nested Steiner triple systems, Discrete Math. 152 (1996), 315–320. [4] Y. Chang, G. Lo Faro and A. Tripodi, The spectrum of Meta(K4 −e > K3 +e, λ) with any λ, Util. Math. 72 (2007), 3–22. [5] Y. Chang, T. Feng, G. Lo Faro and A. Tripodi, The existence spectrum of (3) (3) Meta(K4 > K4 − e), Sci. China Math. 53(11) (2010), 2865–2876. [6] S. K¨ u¸cu ¨k¸cif¸ci and C. C. Lindner, The metamorphosis of λ-fold block designs with block size four into λ-fold kite systems, J. Combin. Math. Combin. Comput. 40 (2002), 241–252. [7] S. K¨ u¸cu ¨k¸cif¸ci and C. C. Lindner, The metamorphosis of λ-fold block designs with block size four into K4 \ e designs, λ ≥ 2, Util. Math. 63 (2003), 239–254. [8] S. K¨ u¸cu ¨k¸cif¸ci, C. C. Lindner, and A. Rosa The metamorphosis of λ-fold block designs with block size four into a maximum packing of λKn with 4-cycles, Discrete Math. 278(1–3) (2004), 175–193. [9] S. K¨ u¸cu ¨k¸cif¸ci, The metamorphosis of λ-fold block designs with block size four into maximum packings of λKn with kites, Util. Math. 68 (2005), 165–195. [10] S. K¨ u¸cu ¨k¸cif¸ci, C. C. Lindner and E. S¸. Yazıcı, The full metamorphosis of λ-fold block designs with block size four into a λ-fold 4-cycle systems, Ars Combin. 104 (2012), 81–96. [11] S. K¨ u¸cu ¨k¸cif¸ci, C. C. Lindner and E. S¸. Yazıcı, The full metamorphosis of λ-fold block designs with block size four into a λ-fold triple systems, Ars Combin. 106 (2012), 337–351. [12] S. K¨ u¸cu ¨k¸cif¸ci, B. Smith and E. S¸. Yazıcı, The full metamorphosis of λ-fold block designs with block size four into a λ-fold kite systems, Util. Math., to appear. [13] C. C. Lindner and A. Street, The metamorphosis of λ-fold block designs with block size four into λ-fold 4-cycle systems, Bulletin of the ICA 28 (2000), 7-18.
204
S. Küçükçifçi / Electronic Notes in Discrete Mathematics 40 (2013) 199–204
[14] C. C. Lindner and A. Rosa, The metamorphosis of λ-fold block designs with block size four into λ-fold triple systems, J. Statist. Plann. Inference 106(1–2) (2002), 69–76. [15] C. C. Lindner and A. Rosa, The metamorphosis of block designs with block size four into (K4 \ e)-designs, Util. Math. 61 (2002), 33–46. [16] C. C. Lindner and A. Tripodi, The metamorphosis of K4 \ e-designs into maximum packings of Kn with 4-cycles, Ars Combin. 75 (2005), 333–349. [17] C. C. Lindner, G. Lo Faro and A. Tripodi, The metamorphosis of λ-fold kite systems into maximum packings of λKn with triangles, J. Combin. Math. Combin. Comput. 56 (2006), 171–189. [18] C. C. Lindner and A. Tripodi, The metamorphosis of λ-fold K4 − e designs into maximum packings of λKn with 4-cycles, J. Statist. Plann. Inference 138(11) (2008), 3316–3325. [19] G. Lo Faro and A. Tripodi, The spectrum of Meta(K3 + e > P4 , λ) and Meta(K3 + e > H4 , λ) with any λ, Discrete Math. 309(2) (2009), 354–362. [20] G. Ragusa, Complete simultaneous metamorphosis of λ-fold kite systems, J. Combin. Math. Combin. Comput. 73 (2010), 159–180. [21] L. Yuan, Y. Zhang, Y. Hou and Q. Kang, The spectrum of M eta(H > T, λ) for all subgraphs H and T of K4 (I), to appear. [22] Y. Zhang, Y. Hou and Q. Kang, The spectrum of M eta(H > T, λ) for all subgraphs H and T of K4 (II), Bulletin of the ICA 59 (2010), 46-70.