ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 10 (2016) 371-381
The spectrum of a-resolvable A-fold (K4 — e)-designs
Mario Gionfriddo *
Dipartimento di Matematica e Informatica, Universitä di Catania, Catania, Italia
Giovanni Lo Faro *
Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universita di Messina, Messina, Italia
Salvatore Milici *
Dipartimento di Matematica e Informatica, Universita di Catania, Catania, Italia
Antoinette Tripodi §
Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universita di Messina, Messina, Italia
Received 20 November 2015, accepted 28 December 2015, published online 1 March 2016
A A-fold G-design is said to be a-resolvable if its blocks can be partitioned into classes such that every class contains each vertex exactly a times. In this paper we study the a-resolvability for A-fold (K4 - e)-designs and prove that the necessary conditions for their existence are also sufficient, without any exception.
Keywords: a-resolvable G-design, a-parallel class, (K4 — e) -design. Math. Subj. Class.: 05B05
*	Supported by PRIN and I.N.D.A.M. (G.N.S.A.G.A.), Italy. tSupportedby PRIN and I.N.D.A.M. (G.N.S.A.G.A.), Italy.
*	Supported by MIUR and I.N.D.A.M. (G.N.S.A.G.A.), Italy. § Supported by PRIN and I.N.D.A.M. (G.N.S.A.G.A.), Italy.
E-mail addresses: gionfriddo@dmi.unict.it (Mario Gionfriddo), lofaro@unime.it (Giovanni Lo Faro), milici@dmi.unict.it (Salvatore Milici), atripodi@unime.it (Antoinette Tripodi)
Abstract
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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1 Introduction
For any graph r, let V(T) and E(T) be the vertex-set and the edge-set of r, respectively, and Ar be the graph r with each of its edges replicated A times. Throughout the paper Kv will denote the complete graph on v vertices, while Kn \ Kh will denote the graph with V(Kn) as vertex-set and E(Kn) \ E(Kh) as edge-set (this graph is sometimes referred to as a complete graph of order n with a hole of size h); finally, Kni,n2,...,nt will denote the complete multipartite graph with t-parts of sizes ni, n2,..., nt.
Let G and H be simple finite graphs. A A-fold G-design of H ((AH, G)-design in short) is a pair (X, B) where X is the vertex-set of H and B is a collection of isomorphic copies (called blocks) of the graph G, whose edges partition the edges of AH. If A = 1, we drop the term "1-fold". If H = Kv, we refer to such a A-fold G-design as one of order v. A (AH, G)-design is balanced if for every vertex x of H the number of blocks containing x is a costant r.
A (AH, G)-design is said to be a-resolvable if it is possible to partition the blocks into classes (often referred to as a-parallel classes) such that every vertex of H appears in exactly a blocks of each class. When a = 1, we simply speak of resolvable design and parallel classes. The existence problem of resolvable G-decompositions has been the subject of an extensive research (see [1,4, 5, 7,8,9, 10, 11, 12, 14, 15, 16, 18, 19,21,24]). The a-resolvability, with a > 1, has been studied for: G = K3 by D. Jungnickel, R. C. Mullin, S. A. Vanstone [13], Y. Zhang and B. Du [25]; G = K4 by M. J. Vasiga, S. Furino and A.C.H. Ling [22]; G = C4 by M.X. Wen and T.Z. Hong [17].
In this paper we investigate the existence of an a-resolvable A-fold (K4 - e)-design (where K4 - e is the complete graph K4 with one edge removed). In what follows, by (a, b, c; d) we will denote the graph K4 - e having {a, b, c, d} as vertex-set and {{a, 6}, {a, c}, {b, c}, {a, d}, {b, d}} as edge-set. Basing on the definitions given above, we can derive the following necessary conditions:
(1)	Av(v - 1) = 0 (mod 10);
(2)	av = 0 (mod 4);
(3)	2A(v - 1) = 0 (mod 5a).
Note that, since the number of a-parallel classes of an a-resolvable A-fold (K4 - e)-design of order v is 2A(5vQ-i) and every vertex appears exactly a times in each of them, we have the following theorem.
Theorem 1.1. Any a-resolvable A-fold (K4 - e)-design is balanced.
From Conditions (1) - (3) we can desume minimum values for a and A, say a0 and A0, respectively. Similarly to Lemmas 2.1, 2.2 in [22], we have the following lemmas.
Lemma 1.2. If an a-resolvable A-fold (K4 - e)-design of order v exists, then a0| a and Ao| A.
Lemma 1.3. If an a-resolvable A-fold (K4 - e)-design of order v exists, then a ta-resolvable nA-fold (K4 - e)-design of order v exists for any positive integers n and t with 11 ^oi).
1 5a
The above two lemmas imply the following theorem (for the proof see Theorem 2.3 in [22]).
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Theorem 1.4. If an a0-resolvable Xo-fold (K4 — e)-design of order v exists and a and A satisfy Conditions (1) — (3), then an a-resolvable A-fold (K4 — e)-design of order v exists.
Therefore, in order to show that the necessary conditions for a-resolvable designs are also sufficient, we simply need to prove the existence of an a0-resolvable A0-fold (K4 — e)-design of order v, for any given v.
2 Auxiliary definitions
A (AKnijn2i...jnt, G)-design is known as a A-fold group divisible design, G-GDD in short, of type {ni, n2,..., nt} (the parts are called the groups of the design). We usually use an "exponential" notation to describe group-types: the group-type 1®2j3f... denotes i occurrences of 1, j occurrences of 2, etc. When G = Kn we will call it an n-GDD.
If the blocks of a A-fold G-GDD can be partitioned into partial a-parallel classes, each of which contains all vertices except those of one group, we refer to the decomposition as a A-fold (a, G)-frame; when a = 1, we simply speak of A-fold G-frame (n-frame if additionally G = Kn). In a A-fold (a, G)-frame the number of partial a-parallel classes missing a specified group of size g is 2a|£(G)|.
An incomplete a-resolvable A-fold G-design of order v + h, h > 1, with a hole of size h is a (A(Kv+h \ Kh), G)-design in which there are two types of classes, A(h—1g(G)G)I
partial classes which cover every vertex a times except those in the hole and 2a|E(G)| full classes which cover every vertex of Kv+h a times.
3 v = 0 (mod 4)
In [4, 5, 23] it was showed that there exists a resolvable (K4 - e)-design of order v = 16 (mod 20); while, for every v = 0,4, 8,12 (mod 20) Gionfriddo et al. ([7]) proved that there exists a resolvable 5-fold (K4 - e)-design of order v. Hence the necessary conditions are also sufficient.
4 v = 1 (mod 2) 4.1 v = 1 (mod 10)
If v = 1 (mod 10), then A0 = 1 and a0 = 4 and so a solution is given by a cyclic (K4 - e)-design ([2]), where every base block generates a 4-parallel class. If v = 10k + 1, k > 4, the desired design can be obtained by developing in Z10k+1 the base blocks listed below:
(1 + 2i, 4k + 1 + i, 1; 2k + 2), i = 3,4,..., f J; (2k + 3 - 2i, 5k + 2 - i, 1; 2k + 2), i = 1, 2,..., [f ]; (1,4k + 1, 3; 6k); (1, 2k + 2, 5; 6k + 1);
where [x\ (or [x]) denote the greatest (or lower) integer that does not exceed (or that exceed) x. If v = 11, 21, 31, the base blocks are: v = 11: (1,10,2; 5) developed in Zn; v = 21: (1,11,3; 15), (1,7, 2; 10) developed in Z21; v = 31: (2,13,1; 5), (1, 27,10; 11), (1,7, 3; 14) developed in Z31.
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4.2 v = 3, 5, 7, 9 (mod 10)
If v = 3, 5, 7,9 (mod 10), then A0 = 5 and a0 = 4 and so a solution is given by a cyclic 5-fold (K4 - e)-design, where every base block generates a 4-parallel class. The required design is obtained by developing in Zv the following blocks:
(1 + i,v - 1 - i, 0; 1), i = 1, 2,..., v—■; (0,1, 2; v - 1).
5 v = 2 (mod 4)
5.1	v = 6 (mod 20)
If v = 6 (mod 20), then A0 = 1 and a0 = 2. In order to prove the existence of a 2-resolvable (K4 - e)-design of order v for every v = 6 (mod 20), preliminarly we need to construct one of order 6.
Lemma 5.1. There exists a 2-resolvable (K4 - e)-design of order 6.
Proof. Let V = {0,1, 2, 3,4,5} be the vertex-set and {(0,1,2; 3), (2, 3,4; 5), (4, 5,0; 1)} be the class.	□
For constructing a 2-resolvable (K4 - e)-design of any order v = 6 (mod 20) and for later use, note that starting from a (K4 - e)-frame of type hn also a A-fold (2, K4 - e)-frame of type hn can be obtained for any A > 0, since necessarily h = 0 (mod 5) and so the number of partial parallel classes missing any group is even.
Lemma 5.2. For every v = 6 (mod 20), there exists a 2-resolvable (K4 - e)-design of order v.
Proof. Let v = 20k + 6. The case k = 0 follows by Lemma 5.1. For k > 0, consider a (2, K4 - e)-frame of type 54fc+1 ([5]) with groups G^ i = 1,2,..., 4k + 1 and a new vertex to. For each i = 1, 2,..., 4k + 1, let Pi the unique partial 2-parallel class which misses the group Gi. Place on Gi U {to} a copy of a 2-resolvable (K4 - e)-design of order 6, which exists by Lemma 5.1, and combine its full class with the partial class Pi so to obtain the desired design.	□
5.2	v = 2,10,14,18 (mod 20)
To prove the existence of an a-resolvable A-fold (K4 - e)-design of order v = 2,10,14,18 (mod 20), with minimum values A0 = 5 and a0 = 2, we will construct some small examples most of which will be used as ingredients in the constructions given by the following theorems.
Theorem 5.3. Let v, g, u, and h be positive integers such that v = gu + h. If there exists
i) a 5-fold (2, K4 - e)-frame of type gu; ii) a 2-resolvable 5-fold (K4 - e)-design of order g;
iii) an incomplete 2-resolvable 5-fold (K4 - e)-design of order g + h with a hole of size h;
then there exists a 2-resolvable 5-fold (K4 - e)-design of order v = gu + h.
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Proof. Take a 5-fold (2, K4 - e)-frame of type gu with groups Gj, i = 1,2,..., u and a set H of size h such taht H n (UU=1Gj) = 0. For j = 1,2,..., g, let Pj,j be the j-th 2-partial class which misses the group Gj. Place on HUG1 a copy D1 of a 2-resolvable 5-fold (K4 -e)-design of order g + h having g + h - 1 classes R1,1, R1,2,...,, R1,g, H1,1, H1,2,..., H1,h-1. For i = 2,3,..., u, place on H U Gj a copy Dj of an incomplete 2-resolvable 5-fold (K4 - e)-design of order g + h with H as hole and having h - 1 partial classes Hj,1, Hj,2,..., Hj,h-1 and g full classes Rj,1, Rj,2,...,, Rj,g. Combine the g partial classes P1,j with the full classes R1,1, R1,2,...,, R1,g of D1 and for i = 2,3,..., u the g partial classes Pj,j of Dj with the full classes Rj,1, Rj,2,..., Rj,g so to obtain gu 2-parallel classes on H U (uu=1Gj). Combine the classes H1,1, H1,2,..., H1,h-1 with the partial classes Hj,1, Hj,2,..., Hj,h-1 so to obtain h - 1 2-parallel classes. The result is a 2-resolvable 5-fold (K4 - e)-design of order gu + h with gu + h -1 2-parallel classes. □
The following lemma gives an input design in the construction of Theorem5.5.
Lemma 5.4. There exists a 2-resolvable 5-fold (K4 - e)-GDD of type 23.
Proof. Let {0, 3}, {1,4} and {2,5} be the groups and consider the following classes: P1 =
{(0, 2,1; 4), (1, 5,0; 3), (3,4, 2; 5)}, P2 = {(3, 5,1; 4), (1, 2, 0; 3), (0, 4, 2; 5)}, P3 = {(0, 5,1; 4), (2,4,0; 3), (1, 3, 2; 5)}, P4 = {(2, 3,1; 4), (4, 5, 0; 3), (0,1, 2; 5)}.	□
Theorem 5.5. Let v, g, m, h and u be positive integers such that v = 2gu + 2m + h. If there exists
i) a 3-frame of type m1gu;
ii) a 2-resolvable 5-fold (K4 - e)-design of order 2m + h;
iii) an incomplete 2-resolvable 5-fold (K4 - e)-design of order 2g + h with a hole of size h;
then there exists a 2-resolvable 5-fold (K4 - e)-design of order 2gu + 2m + h.
Proof. Let F be a 3-frame with one group G of cardinality m and u groups Gj, i = 1,2,..., u of cardinality g; such a frame has y partial classes which miss G, each containing gu triples, and, for i = 1,2,..., u, 2 partial classes which miss Gj, each containing
g("-31)+m triples. Expand each vertex 2 times and add a set H of h new vertices. Place on H U (G x{1, 2}) a copy D of a 2-resolvable 5-fold (K4 - e) -design of order 2m + h having 2m + h - 1 classes R1, R2,..., R2m, H1, H2,..., Hh-1. For each i = 1,2,..., u place on H U (Gj x {1,2}) a copy Dj of an incomplete 2-resolvable 5-fold (K4 - e)-design of order 2g + h with H as hole and having h - 1 partial classes Hj,j with j = 1, 2,..., h - 1 and 2g full classes Rj,i, t = 1, 2,..., 2g. For each block b = {x, y, z} of a given class of F place on b x {1, 2} a copy of a 2-resolvable 5-fold (K4 - e)-GDD of type 23 from Lemma 5.4, having {x1, x2}, {y1, y2} and {z1, z2} as groups. This gives 2m partial classes (whose blocks are copies of K4 - e) which miss G x {1,2} and 2g partial classes which miss Gj x {1,2}, i =1,2,..., u. Combine the 2m partial classes which miss the group G x {1,2} with the classes R1, R2,..., R2m so to obtain 2m classes. For i = 1,2,..., u combine the 2g partial classes which miss the group Gj x {1, 2} with the full classes of Dj so to obtain 2gu classes. Finally, combine the h - 1 classes H1, H2,..., Hh-1 of D with the partial classes of Dj so to obtain h - 1 classes. This gives a 2-resolvable 5-fold (K4 - e)-design of order v and v - 1 2-parallel classes.	□
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Theorem 5.6. Let v, k and h be non-negative integers. If there exists
i) an incomplete a-resolvable X-fold (K4 — e)-design of order v + k + h with a hole of size k + h;
ii) an incomplete a-resolvable X-fold (K4 — e) -design of order k + h with a hole of size h;
then there exists an incomplete a-resolvable X-fold (K4 — e)-design of order v + k + h with a hole of size h.
Lemma 5.7. There exists a resolvable (K4 — e )-GDD of type 52101.
Proof. Let Z10 U {to0, to^ ..., to9} be the vertex-set and 2Z10, 2Z10 + 1, {to0, to^ ..., to9} be the groups. The desired design is obtained by adding 2 (mod 10) to the following base blocks, including the subscripts of to: (0,1, to0; to1), (2, 5, to0; to1), (4,9, to0; to1), (6,3, to0; to1), (8,7, to0; to1). The parallel classes are generate by every base block.	□
Lemma 5.8. There exists a 2-resolvable 5-fold (K4 — e)-GDD of type 103.
Proof. Start with the 2-resolvable 5-fold (K4 — e)-GDD G of type 23 of Lemma 5.4 with groups Gi, i = 1, 2,3. For each block b = (x, y, z; t) of a given 2-parallel class of G consider a copy of a resolvable (K4 — e)-GDD of type 52101 where {x} x Z5, {y} x Z5, {z, t} x Z5 are the groups.	□
Lemma 5.9. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 6 with a hole of size 2.
Proof. On V = Z4 U H, where H = {to1, to2} is the hole, consider the partial class
{(1,3,0; 2), (0,2,1; 3)} and the four full classes obtained by developing {(0,2, to1; to2), (to1, 1, 0; 3), (to2, 2, 3; 1)} in Z4, where TOi + 1 = TOi for i = 1, 2.	□
Lemma 5.10. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 10 with a hole of size 2.
Proof. On V = Z8 U H, where H = {to1, to2} is the hole, consider the partial class
{(0,4,2; 6), (1,5,3; 7), (2,6,4; 0), (3,7, 5; 1)} and the eight full classes obtained by developing {(0,1, to1; 3), (2, 3, to2; 7), (to1, 5, 6; 2), (to2, 6,4; 5), (4, 7,1; 0)} in Z8, where TOi + 1 = TOi for i = 1,2.	□
Lemma 5.11. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 14 with a hole of size 4.
Proof. Let V = Z10 U H be the vertex-set, where H = {to1, to2, to3, to4} is the hole. The partial classes are obtained by adding 2 (mod 10) to the base blocks (2,6,9; 5), (5,9, 2; 8), (8, 7,6; 9), each block generating a partial class; while, the full classes are obtained by adding 2 (mod 10) to the following base blocks partitioned into two full classes, each class generating five full classes: {(0,8, to1; to2), (1,5, to3; to4), (to1, 4, 0; 9), (to2, 6, 2; 3), (to3, 3, 7; 8), (TO4, 9,1;4), (2, 7, 6; 5)}, {(1, 5, TO1; TO2), (0, 8, to3; to4), (to 1, 3, 9; 4), (to2, 9, 7;0), (to3, 2, 6; 1), (to4, 6, 8; 3), (4, 7, 2; 5)}, where TOi + 1^oofor i = 1,2, 3,4.	□
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Lemma 5.12. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 14 with a hole of size 2.
Proof. On V = Z12 U H, where H = {to^ to2} is the hole, consider the partial class
{(0, 6,3; 9), (1,7,4; 10), (2, 8,5; 11), (3, 9,6; 0), (4,10, 7; 1), (5,11, 8; 2)} and the twelve full classes obtained by developing {(0,1, to^ 11), (2,4, to2; 10), (to1, 10,6; 5), (to2, 9, 2; 0), (3, 7,8; 1), (5, 8,7; 9), (6,11, 3; 4)} in Z12, where to, + 1 = to, for i = 1,2. □
Lemma 5.13. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 22 with a hole of size 6.
Proof. Let V = Z16 U H be the vertex-set, where H = {to1, to2, ..., to6} is the hole. In Zi6 develop the full 2-parallel base class {(0,3, toi; 12), (1,5, TO2;2), (8,13, to3;4), (14, 15, to4; 11), (6,11, to5; to6 ), (to1, 2,1;3), (to2, 4,13; 8), (to3, 7, 0; 14), (to4, 9, 6; 10), (to5, 10, 5; 15), (to6, 12,7; 9)}. Additionally, include the partial 2-parallel class {(0,8, 2; 10), (1, 9, 3; 11), (2,10,4; 12), (3,11, 5; 13), (4,12, 6; 14), (5,13, 7; 15), (6,14, 8; 0), (7, 15, 9;1)} repeated five times.	□
As consequence of Lemmas 5.9 and 5.13, by Theorem 5.6 the following lemma follows.
Lemma 5.14. There exists a 2-resolvable 5-fold (K4 — e)-design of order 22 with a hole of size 2.
Lemma 5.15. There exists a 2-resolvable 5-fold (K4 — e)-design of order 10.
Proof. Let V = Z9 U {to} be the vertex-set. The required design is obtained by developing the base class {(to, 0,6; 5), (1,5,4; 3), (7,8,1; to), (2, 6, 7; 8), (3,4, 2; 0)} in Z9. □
Lemma 5.16. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 30 with a hole of size 10.
Proof. Start from a 2-resolvable 5-fold (K4 — e)-GDD of type 103 (which exists by Lemma 5.8) having G,, i = 1, 2, 3, as groups. Fill in the groups G2 and G3 with a copy of a 2-resolvable 5-fold (K4 — e)-design of order 10, which exists by Lemma 5.15. This gives an incomplete 2-resolvable 5-fold (K4 — e)-design of order 30 with G1 as hole.	□
Lemma 5.17. There exists an incomplete 2-resolvable 5-fold (K4 — e)-design of order 38 with a hole of size 12.
Proof. Let V = Z26 U H be the vertex-set, where H = {to1, to2, ..., to12} is the hole. The partial classes are: {(i, 13 + i, 2 + i; 15 + i) : i = 0,1,..., 12}, repaeated five times; {(2i, 10 + 2i, 3 + 2i; 7 + 2i) : i = 0,1,..., 12} and {(1 + 2i, 11 + 2i, 4 + 2i; 8 + 2i) : i = 0,1,..., 12}, repeated twice; {(2i, 10+2i, 1 + 2i; 9+2i) : i = 0,1,..., 12}; {(1 + 2i, 11 + 2i, 2 + 2i; 10 + 2i) : i = 0,1,..., 12}. The full classes are obtained by developing in V = Z26 the full base class {(toi, 2,1;7), (to2, 12,3; 24), (to3, 16,4; 11), (to4, 13,5; 25), (to5, 15, 9; 22), (to6, 17,11; 23), (to7, 19,18; 20), (to8, 14,10; 18), (to9, 4, 0; 8), (to10, 9,17; 19), (to 11, 7, 2; 12), (to12, 15, 3; 24), (1, 5, to1; to2), (10, 20, to3; to4), (6, 23, to5; to6), (16, 21, to7; to8), (22, 25, to9; to10), (13, 21, to11; to12), (0,14, 6; 8)}.	□
As consequence of the existence of a 2-resolvable 5-fold (K4 — e)-design of order v = 4,12 (see Section 3 and Theorem 1.4) and Lemmas 5.1, 5.11, 5.13, 5.16, 5.17, 5.15, by Theorem 5.6 the following lemma follows.
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Lemma 5.18. There exists a 2-resolvable 5-fold (K4-e)-design of order v = 14,22,30,38.
Lemma 5.19. There exists a 2-resolvable 5-fold (K4 — e)-design of order v = 42,234.
Proof. Start with a resolvable 3-GDD of type 36 ([20]). Expand each vertex 2 times and for each triple b of a given parallel class place on b x {1, 2} a copy of a 2-resolvable 5-fold (K4 — e)-GDD of type 23, which exists by Lemma 5.4. Finally, fill each group of size 6 with a copy of a 2-resolvable 5-fold (K4 — e)-design of order 6, which exists by Lemma 5.1.	□
Lemma 5.20. There exists a 2-resolvable 5-fold (K4 — e)-design of order v = 50,62.
v-2
Proof. Start from a 3-frame of type 6([3]) and apply Contraction 5.5 with m = g = 6, h = 2 and u = to obtain a 2-resolvable 5-fold (K4 — e)-design of order v = 50,62 (the input designs are: a 2-resolvable 5-fold (K4 — e)-design of order 14, which exists by Lemma 5.18; a 2-resolvable 5-fold (K4 — e)-GDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 — e)-design of order 14 with a hole of size 2, which exists by Lemma 5.12).	□
Lemma 5.21. There exists a 2-resolvable 5-fold (K4 — e)-design of order v = 34,274.
v-2
Proof. Start from a 3-frame of type 4([3]) and apply Theorem 5.5 with m = g = 4, h = 2 and u = ^^g10 to obtain a 2-resolvable 5-fold (K4 — e)-design of order v = 34, 274 (the input designs are: a 2-resolvable 5-fold (K4 — e)-design of order 10, which exists by Lemma 5.15; a 2-resolvable 5-fold (K4 — e)-GDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 — e)-design of order 10 with a hole of size 2, which exists by Lemma 5.10).	□
Lemma 5.22. There exists a 2-resolvable 5-fold (K4 — e)-design of order 70.
Proof. Start from a 3-frame of type 84 ([3]) and apply Theorem 5.5 with m = g = 8, h = 6 and u = 3 to obtain a 2-resolvable 5-fold (K4 — e)-design of order 70 (the input designs are; a 2-resolvable 5-fold (K4 — e)-design of order 22, which exists by Lemma 5.18; a 2-resolvable 5-fold (K4 — e)-RGDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 — e)-design of order 22 with a hole of size 6, which exists by Lemma 5.13).	□
Lemma 5.23. For every v = 2 (mod 20), there exists a 2-resolvable 5-fold (K4 — e)-design of order v.
Proof. Let v=20k + 2. The case v = 22,42,62 are covered by Lemmas 5.18, 5.19 and 5.20. For k > 4, start from a 5-fold (2, K4 — e)-frame of type 20k ([5]) and apply Theorem 5.3 with h = 2 to obtain a 2-resolvable 5-fold (K4 — e)-design of order v (the input designs are a 2-resolvable 5-fold (K4 — e)-design of order 22, which exists by Lemma 5.18, and an incomplete 2-resolvable 5-fold (K4 — e)-design of order 22 with a hole of size 2, which exists by Lemma 5.14).	□
Lemma 5.24. For every v = 10 (mod 20), there exists a 2-resolvable 5-fold (K4 — e)-design of order v.
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Proof. Let v=20k + 10. The case v = 10, 30, 50, 70 are covered by Lemmas 5.15, 5.18, 5.20 and 5.22. For k > 4, start from a 5-fold (2, K4 - e)-fTame of type 20k ([5]) and apply Theorem 5.3 with g = 20 and h =10 to obtain a 2-resolvable 5-fold (K4 - e)-design of order v (the input designs are a 2-resolvable 5-fold (K4 - e)-design of order 10, which exists by Lemma 5.15, and an incomplete 2-resolvable 5-fold (K4 - e)-design of order 30 with a hole of size 10, which exists by Lemma 5.16).	□
Lemma 5.25. For every v = 14 (mod 20), there exists a 2-resolvable 5-fold (K4 - e)-design of order v.
Proof. Let v=20k + 14. The case v = 14, 34, 234,274 are covered by Lemmas 5.18, 5.19 and 5.21. For k > 2, k / {11,13}, start from a 5-fold (2, K4 - e)-frame of type 102fc+1 ([5]), apply Theorem 5.3 with h = 4 and proceed as in Lemma 5.24.	□
Lemma 5.26. For every v = 18 (mod 60), there exists a 2-resolvable 5-fold (K4 - e)-design of order v.
Proof. Let v=60k + 18. Take a resolvable 3-GDD of type 310k+3 ([6]). Expand each vertex 2 times and for each block b of a parallel class place on b x {1, 2} a copy of a 2-resolvable 5-fold (K4 - e)-GDD of type 23 which exists by Lemma 5.4, so to obtain a 2-resolvable 5-fold (K4 - e)-GDD of type 610k+3. Finally, fill in each group of size 6 with a copy of a 2-resolvable 5-fold (K4 - e)-design, which exists by Lemma 5.1.	□
Lemma 5.27. For every v = 38 (mod 60), there exists a 2-resolvable 5-fold (K4 - e)-design of order v.
Proof. Let v = 60k + 38. The case v = 38 follows by Lemmas 5.18. For k > 1, start from a 3-fTame of type 65k+3 ([6]) and apply Theorem 5.5 with m = g = 6, h = 2 and u = 5k + 2 to obtain a 2-resolvable 5-fold (K4 - e)-design of order v (the input designs are: a 2-resolvable 5-fold (K4 - e)-design of order 14, which exists by Lemma 5.18; a 2-resolvable 5-fold (K4 - e)-GDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 - e)-design of order 14 with a hole of size 2, which exists by Lemma 5.11)	□
Lemma 5.28. For every v = 58 (mod 120), there exists a 2-resolvable 5-fold (K4 - e)-design of order v.
Proof. Let v = 120k + 58. Start from a 3-frame of type 415k+7 ([6]) and apply Theorem 5.5 with m = g = 4, h = 2 and u = 15k + 6 to obtain a 2-resolvable 5-fold (K4 - e)-design of order v (the input designs are: a 2-resolvable (K4 - e)-design of order 10, which exists by Lemma 5.15; a 2-resolvable 5-fold (K4 - e)-RGDD of type 23, which exists by Lemma 5.4; an incomplete 2-resolvable 5-fold (K4 - e)-design of order 10 with a hole of size 2, which exists by Lemma 5.10).	□
Lemma 5.29. For every v = 118 (mod 120), there exists a 2-resolvable 5-fold (K4 - e)-design of order v.
Proof. Let v = 120k + 118. Start from a 3-frame of type 101415k+12, k > 0, ([6]) and apply Theorem 5.5 with h = 2 to obtain a 2-resolvable 5-fold (K4 - e)-design of order v (the input designs are: a 2-resolvable 5-fold (K4 - e)-design of order 22, which exists by Lemma 5.18; a 2-resolvable 5-fold (K4 - e)-RGDD of type 23, which exists by Lemma
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Ars Math. Contemp. 10 (2016) 255-268
5.4; an incomplete 2-resolvable 5-fold (K4 — e)-design of order 10 with a hole of size 2,
which exists by Lemma 5.10).	□
6 Main result
The results obtained in the previous sections can be summarized into the following theorem.
Theorem 6.1. The necessary conditions (1) — (3) for the existence of a-resolvable X-fold
(K4 — e)-designs are also sufficient.
References
[1]	J.-C. Bermond, K. Heinrich and M.-L. Yu, Existence of resolvable path designs, European J. Combin. 11 (1990), 205-211, doi:10.1016/S0195-6698(13)80120-5, http://dx.doi. org/10.1016/S0195-6698(13)80120-5.
[2]	J.-C. Bermond and J. Schonheim, G-decomposition of Kn, where G has four vertices or less, Discrete Math 19 (1977), 113-120.
[3]	C. J. Colbourn and J. H. Dinitz (eds.), Handbook of combinatorial designs, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2nd edition, 2007.
[4]	C. J. Colbourn, D. R. Stinson and L. Zhu, More frames with block size four, J. Combin. Math. Combin. Comput. 23 (1997), 3-19.
[5]	G. Ge and A. C. H. Ling, On the existence of resolvable K4 — e designs, J. Combin. Des. 15 (2007), 502-510, doi:10.1002/jcd.20155, http://dx.doi.org/10.10 02/ jcd.20155.
[6]	G. Ge, R. Rees and N. Shalaby, Kirkman frames having hole type h^m1 for small h, Des. Codes Cryptogr. 45 (2007), 157-184, doi:10.1007/s10623-007-9105-2, http://dx.doi. org/10.1007/s10623-007-9105-2.
[7]	M. Gionfriddo, G. Lo Faro, S. Milici and A. Tripodi, Resolvable (kv — e) -designs of order v and index A (to appear).
[8]	M. Gionfriddo and S. Milici, On the existence of uniformly resolvable decompositions of Kv and Kv — I into paths and kites, Discrete Math. 313 (2013), 2830-2834, doi:10.1016/j.disc. 2013.08.023, http://dx.doi.org/10.1016/j-disc.2013.08.023.
[9]	M. Gionfriddo and S. Milici, Uniformly resolvable H-designs with H = {P-3,P4}, Aus-tralas. J. Combin. 60 (2014), 325-332.
[10]	H. Hanani, On resolvable balanced incomplete block designs, J. Combinatorial Theory Ser. A 17 (1974), 275-289.
[11]	H. Hanani, D. K. Ray-Chaudhuri and R. M. Wilson, On resolvable designs, Discrete Math. 3 (1972), 343-357.
[12]	J. D. Horton, Resolvable path designs, J. Combin. Theory Ser. A 39 (1985), 117-131, doi: 10.1016/0097-3165(85)90033-0, http://dx.doi.org/10.1016/00 97-3165(85) 90033-0.
[13]	D. Jungnickel, R. C. Mullin and S. A. Vanstone, The spectrum of a-resolvable block designs with block size 3, Discrete Math. 97 (1991), 269-277, doi:10.1016/0012-365X(91)90443-6, http://dx.doi.org/10.1016/0012-365X(91)9044 3-6.
[14]	S. Kucukcifci, G. Lo Faro, S. Milici and A. Tripodi, Resolvable 3-star designs, Discrete Math. 338 (2015), 608-614, doi:10.1016/j.disc.2014.11.013, http://dx.doi.org/10.1016/ j.disc.2014.11.013.
M. Giongriddo, et al.: The spectrum of a-resolvable X-fold (K4 — e)-designs
381
[15]	S. Kiicukgifci, S. Milici and Z. Tuza, Maximum uniformly resolvable decompositions of Kv and Kv — I into 3-stars and 3-cycles, Discrete Math. 338 (2015), 1667-1673, doi:10.1016/j. disc.2014.05.016, http://dx.doi.org/10.1016/j.disc.2 014.05.016.
[16]	G. Lo Faro, S. Milici and A. Tripodi, Uniformly resolvable decompositions of Kv into paths on two, three and four vertices, Discrete Math. 338 (2015), 2212-2219, doi:10.1016/j.disc.2015. 05.030, http://dx.doi.org/10.1016/j.disc.2015.05.030.
[17]	X. W. Ma and Z. H. Tian, a-resolvable cycle systems for cycle length 4, J. Math. Res. Exposition 29 (2009), 1102-1106.
[18]	S. Milici, A note on uniformly resolvable decompositions of Kv and Kv — I into 2-stars and 4-cycles, Australas. J. Combin. 56 (2013), 195-200.
[19]	S. Milici and Z. Tuza, Uniformly resolvable decompositions of Kv into P3 and K3 graphs, Discrete Math. 331 (2014), 137-141, doi:10.1016/j.disc.2014.05.010, http://dx.doi.org/ 10.1016/j.disc.2014.05.010.
[20]	R. Rees and D. R. Stinson, On resolvable group-divisible designs with block size 3, Ars Combin. 23 (1987), 107-120.
[21]	R. Su and L. Wang, Minimum resolvable coverings of Kv with copies of K4 — e, Graphs Combin. 27 (2011), 883-896, doi:10.1007/s00373-010-1003-0, http://dx.doi.org/ 10.1007/s00373-010-1003-0.
[22]	T. M. J. Vasiga, S. Furino and A. C. H. Ling, The spectrum of a-resolvable designs with block size four, J. Combin. Des. 9 (2001), 1-16, doi:10.1002/1520-6610(2001) 9:1(1::AID-JCD1)3.0.C0;2-H, http://dx.doi.org/10.1002/1520-6610(2001) 9:1<1::AID-JCD1> 3.0.CO;2-H.
[23]	L. Wang, Completing the spectrum of resolvable (K4 — e)-designs, Ars Combin. 105 (2012), 289-291.
[24]	L. Wang and R. Su, On the existence of maximum resolvable (K4 — e)-packings, Discrete Math. 310 (2010), 887-896, doi:10.1016/j.disc.2009.10.007, http://dx.doi.org/10. 1016/j.disc.2009.10.007.
[25]	Y. Zhang and B. Du, a-resolvable group divisible designs with block size three, J. Combin. Des. 13 (2005), 139-151, doi:10.1002/jcd.20028, http://dx.doi.org/10.10 02/ jcd.20028.