ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 16 (2019) 119-139
https://doi.org/10.26493/1855-3974.1490.eea
(Also available at http://amc-journal.eu)
The Doyen-Wilson theorem for 3-sun systems*
Giovanni Lo Faro , Antoinette Tripodi
Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universita di Messina, Messina, Italia
Received 25 September 2017, accepted 18 April 2018, published online 20 September 2018
A solution to the existence problem of G-designs with given subdesigns is known when G is a triangle with p = 0,1, or 2 disjoint pendent edges: for p = 0, it is due to Doyen and Wilson, the first to pose such a problem for Steiner triple systems; for p = 1 and p = 2, the corresponding designs are kite systems and bull designs, respectively. Here, a complete solution to the problem is given in the remaining case where G is a 3-sun, i.e. a graph on six vertices consisting of a triangle with three pendent edges which form a 1-factor.
Keywords: 3-sun systems, embedding, difference set. Math. Subj. Class.: 05B05, 05B30
1 Introduction
If G is a graph, then let V(G) and E(G) be the vertex-set and edge-set of G, respectively. The graph Kn denotes the complete graph on n vertices. The graph Km \ Kn has vertex-set V(Km) containing a distinguished subset H of size n; the edge-set of Km \ Kn is E(Km) but with the edges between the n distinguished vertices of H removed. This graph is sometimes referred to as a complete graph of order m with a hole of size n.
Let G and r be finite graphs. A G-design of r is a pair (X, B) where X = V(r) and B is a collection of isomorphic copies of G (blocks), whose edges partition E(r). If r = Kn, then we refer to such a design as a G-design of order n.
A G-design (Xi, Bi) of order n is said to be embedded in a G-design (X2, B2) of order m provided X1 C X2 and B1 C B2 (we also say that (X1, B1) is a subdesign (or subsystem) of (X2, B2) or (X2, B2) contains (X1, B1) as subdesign). Let N(G) denote the set of integers n such that there exists a G-design of order n. A natural question to ask is: given n, m G N(G), with m > n, and a G-design (X, B) of order n, does exists a G-design of order m containing (X, B) as subdesign? Doyen and Wilson were the first to
* Supported by I.N.D.A.M. (G.N.S.A.G.A.).
E-mail addresses: lofaro@unime.it (Giovanni Lo Faro), atripodi@unime.it (Antoinette Tripodi)
Abstract
©® This work is licensed under https://creativecommons.org/licenses/by/4.0/
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pose this problem for G = K3 (Steiner triple systems) and in 1973 they showed that given n, m G N (K3) = {v : v = 1,3 (mod 6)}, then any Steiner triple system of order n can be embedded in a Steiner triple system of order m if and only if m > 2n +1 or m = n (see
[3]).	Over the years, any such problem has come to be called a "Doyen-Wilson problem" and any solution a "Doyen-Wilson type theorem". The work along these lines is extensive ([1, 4, 5, 6, 7, 8, 9, 10, 13]) and the interested reader is referred to [2] for a history of this problem.
In particular, taking into consideration the case where G is a triangle with p = 0,1, 2, or 3 mutually disjoint pendent edges, a solution to the Doyen-Wilson problem is known when p = 0 (Steiner triple systems, [3]), p =1 (kite systems, [9, 10]) and p = 2 (bull designs,
[4]).	Here, we deal with the remaining case (p = 3) where G is a 3-sun, i.e. a graph on six vertices consisting of a triangle with three pendent edges which form a 1-factor, by giving a complete solution to the Doyen-Wilson problem for G-designs where G is a 3-sun (3-sun systems).
2 Notation and basic lemmas
The 3-sun consisting of the triangle (a, b, c) and the three mutually disjoint pendent edges {a, d}, {b, e}, {c, f} is denoted by (a, b, c; d, e, f). A 3-sun system of order n (briefly, 3SS(n)) exists if and only if n = 0,1,4, 9 (mod 12) and if (X, S) is a 3SS(n), then |S| = (see [14]).
Let n, m = 0,1,4,9 (mod 12), with m = u + n, u > 0. The Doyen-Wilson problem for 3-sun systems is equivalent to the existence problem of decompositions of K„+n \ Kn into 3-suns.
Let r and s be integers with r < s, define [r, s] = {r, r + 1,..., s} and [s, r] = 0. Let
= [0, u — 1] and H = {ro^ ro2, ..., rot}, H fl = 0. If S = (a, b, c; d, e, f) is a 3-sun whose vertices belong to U H and i G Zu, let S + i = (a + i, b + i, c + i; d + i, e + i, f + i), where the sums are modulo u and ro + i = ro, for every ro G H. The set (S) = {S + i : i G Zu} is called the orbit of S under and S is a base block of (S).
To solve the Doyen-Wilson problem for 3-sun systems we use the difference method (see [11, 12]). For every pair of distinct elements i, j G Zu, define |i — j|u = min{|i — j|,u — |i — j|} and set D„ = {|i — j|„ : i, j G Z„} = {1, 2,..., |_fj}. The elements of are called differences of . For any d G Du, d = u, we can form a single 2-factor {{i, d + i} : i G Zu}, while if u is even and d = u, then we can form a 1-factor {{i, i + u} : 0 < i < u — 1}. It is also worth remarking that 2-factors obtained from distinct differences are disjoint from each other and from the 1-factor.
If D C D„, denote by (Z„ U H, D) the graph with vertex-set V = Z„ U H and the edge-set E = {{i, j} : |i — j|u = d, d G D} U {{ro, i} : ro G H, i G Zu}. The graph (Zu U H, Du) is the complete graph K„+i \ Kt based on U H and having H as a hole. The elements of H are called infinity points.
Let X be a set of size n = 0,1,4, 9 (mod 12). The aim of the paper is to decompose the graph (Zu U X, Du) into 3-suns. To obtain our main result the (Zu U X, Du) will be regarded as a union of suitable edge-disjoint subgraphs of type (Zu U H, D) (where H C X may be empty, while D C Du is always non empty) and then each subgraph will be decomposed into 3-suns by using the lemmas given in this section. From here on suppose u = 0,1, 3, 4, 5, 7, 8, 9,11 (mod 12).
Lemmas 2.1-2.4 give decompositions of subgraphs of type (Zu U H, D) where D
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121
contains particular differences, more precisely, D = {2}, D = {2,4} or D = {1, u}.
Lemma 2.1. Let u = 0 (mod 4), u > 8. Then the graph (Z„ U {toi, to2}, {2}} can be decomposed into 3-suns.
Proof. Consider the 3-suns
(toi, 2 + 4«, 4«; 3 + 4«, 4 + 4«, to2), (to2, 3 + 4«, 1 + 4«; 2 + 4«, 5 + 4«, toi),
for « = 0,1,..., u - 1.	□
Lemma 2.2. Let u = 0 (mod 12). Then the graph (Z„ U {toi, to2, to3, to4}, {2}} can be decomposed into 3-suns.
Proof. Consider the 3-suns
(toi, 12«, 2 + 12«; 7 + 12«,, to3, to4), (toi, 4+ 12«, 6 + 12«; 9 + 12«, to3, to4), (toi, 8 + 12«, 10 + 12«; 11 + 12«, to3, to4), (to2, 2 + 12«, 4 + 12«; 1 + 12«, to3, to4), (to2, 6 + 12«, 8 + 12«; 7 + 12«, to3, to4), (to2, 10 + 12«, 12 + 12«; 11 + 12«, to3, to4), (to3, 1 + 12«, 3 + 12«; 9 + 12«, toi, to2), (to3, 5 + 12«, 7 + 12«; 11 + 12«, toi, 9 + 12«), (to4, 3 + 12«, 5 + 12«; 1 + 12«, toi, to2), (to4, 9 + 12«, 11 + 12«; 7 + 12«, to2, 13 + 12«),
for « = 0,1,..., 12 - 1.	□
Lemma 2.3. The graph (Z„ U {toi, to2, to3, to4}, {2,4}}, u > 7, u = 8, can be decomposed into 3-suns.
Proof. Let u = 4k + r, with r = 0,1,3, and consider the 3-suns
(toi, 4 + 4«, 6 + 4«; 5 + 4«, 8 + 4«, to4), (to2, 5 + 4«, 7 + 4«; 6 + 4«, 9 + 4«, toi), (to3, 6 + 4«, 8 + 4«; 7 + 4«, 10 + 4«, to2), (to4, 7 + 4«, 9 + 4«; 8 + 4«, 11 + 4«, to3),
for « = 0,1,..., k - 3, k > 3, plus the following blocks as the case may be.
If r = 0,
(toi, 0, 2; 1, 4, to4), (to2, 1, 3; 2, 5, toi), (to3, 2,4; 3, 6, TO2), (to4, 3, 5; 4, 7, to3), (toi, 4k - 4, 4k - 2; 4k - 3,0, to4),
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If r = 1,
If r = 3,
(to2, 4k - 3,4k - 1; 4k - 2,1, toi), (to3, 4k - 2,0; 4k - 1, 2, to2), (to4, 4k - 1,1; 0, 3, to3).
(toi, 0, 2; 1, 4, to2),
(to2, 1, 3; 0, 5, toi),
(to3, 2,4; 3, 6, to2),
(to4, 3, 5; 4, 7, to3),
(toi, 4k - 4, 4k - 2; 4k - 3, 4k, to2),
(to2, 4k - 3, 4k - 1; 4k, 0, tox),
(to3, 4k - 2, 4k; 4k - 1,1, toi),
(to4, 4k - 1, 0; 4k - 2, 2, to3),
(to4, 4k, 1; 2, 3, to3).
(toi, 0, 2; 1, 4, to4),
(to2, 1, 3; 2, 5, toi),
(TO3, 2,4; 3, 6, TO2),
(to4, 3, 5; 4, 7, to3),
(toi, 4k - 4, 4k - 2; 4k - 3,4k, to4),
(to2, 4k - 3, 4k - 1; 4k - 2,4k + 1, toi),
(to3, 4k - 2, 4k; 4k - 1, 4k + 2, to2),
(to4, 4k - 1, 4k + 1; 4k, 0, to3),
(toi, 4k, 4k + 2; 4k + 1,1, to4),
(to2, 4k + 1, 0; 4k + 2, 2, to4),
(to3, 4k + 2,1; 0, 3, to4).
With regard to the difference 4 in Z7, note that |4|7 obtained for k = 1 and r into 3-suns.
3 and the seven distinct blocks
3 gives a decomposition of (Z7 U {toi, to2, to3, to4}, {2,3}}
□
Lemma 2.4. Let u = 0 (mod 3), u > 12. Then the graph (Zu U {toi, to2, ..., to8}, {1, U }} can be decomposed into 3-suns.
Proof. If u = 0 (mod 6) consider the 3-suns:
(toi, 2i, u + 2i; 2u + 2i, to5, to6), i = 0,1,
u - 1
6 1,
u _ 1
6 1,
(toi, 1 + 2i, u + 1 + 2i; 2u + 1 + 2i, to6, to5), i = 0,1,.. (to2, 2u + 2i, u + 2i; 2 + 2i, 2i, to5), i = 0,1,..., u - 2, (to2, 2u + 1 + 2i, u + 1 + 2i; 3 + 2i, 1 + 2i, to6), i = 0,1,..., u - 2, (to2, u - 2, 2u - 2; 0, u - 2, to5),
G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems	121
- - 1
6 1
to2,u - 1, 2- - 1; 1, - - 1, to6), to3, 2i, 1 + 2i; 2- + 2i, to7, to8), i = 0,1,..., - - 1, TO3, - + 2i, - + 1 + 2i; 2- + 1 + 2i, TO7, tos), i = 0,1, to4, 1 + 2i, 2 + 2i; 2- + 2 + 2i, to7, to8), i = 0,1,..., - - 1, to4, - + 1 + 2i, - + 2 + 2i; 2- + 1 + 2i, to7, to8), i = 0,1,..., - - 1, to5, 2- + 2i, 2- + 1 + 2i; 1 + 2i, to7, to8), i = 0,1,..., - - 1, to6, 2- + 3 + 2i, 2- + 4 + 2i; 2 + 2i, to7, to8), i = 0,1,..., - - 2, (to6, 2- + 1, 2- +2; 2-, to7, to8).
If u = 3 (mod 6) consider the 3-suns:
(toi, 2i, - + 2i; 2- + 2i, to5, to6), i = 0,1,..., ,
TOi
TO2 TO2 TO2 TO2 TO3 TO3 TO3 TO3
to4 to4
TO5 TO6 TO6 to7
m-9 6 ,
1 + 2i, - + 1 + 2i; 2- + 1 + 2i, to6, to5), i = 0,1, 2- + 2i, - + 2i; 2 + 2i, 2i, to5), i = 0,1,..., , u - 1, 2- - 1; 0, - - 1, to5),
2- + 1 + 2i, - + 1 + 2i; 3 + 2i, 1 + 2i, to6), i = 0,1,..., ,
u - 2, 2- - 2; 1, - - 2, to6),
2i, 1 + 2i; 2- + 2i, to7, to8), i = 2, 3,..., ,
0,1; 2-, to6, TO8),
2, 3; 2- + 2, TO6, tos),
-	+ 1 + 2i, - + 2 + 2i; 2- + 1 + 2i, to7, to8), i = 0,1,... 1 + 2i, 2 + 2i; 2- + 2 + 2i, to7, to8), i = 0,1,..., ,
-	+ 2i, - + 1 + 2i; 2- + 1 + 2i, to7, to8), i = 0,1,..., , 2- + 2i, 2- + 1 + 2i; 1 + 2i, to7, to8), i = 0,1,..., ,
2- + 1 + 2i, 2- + 2 + 2i; 4 + 2i, to7, to8), i = 0,1,..., ^g15, u - 2,u - 1; 2-, to7, to8), u - 1,0; 2, to5, to8).
m-9 6 ,
□
Lemmas 2.5 - 2.9 allow to decompose (Zm U H, D) where u is even and D contains the
difference -.
Lemma 2.5. Let u be even, u > 8. Then the graph (Zm U {to1, to2, to3}, {1, -}) can be decomposed into 3-suns.
Proof. Consider the 3-suns
(to1, 2i, 1 + 2i; - + 2 + 2i, - + 2i, to3), i = 0,1,
(toi, - - 2, - - 1; -, u - 2, TO3),
(to2, 1 + 2i, - + 1 + 2i; 2i, 2 + 2i, to1), i = 0,1,
(TO3, - + 1 + 2i, - + 2i; 2i, - + 2 + 2i, TO2), i =
-
4 - 2,
- - 1
4 1,
0,1,...,- - 1.
□
Lemma 2.6. Let u = 0 (mod 12). Then the graph (Zm U {to1, to2, to3, to4}, {1, -}) can be decomposed into 3-suns.
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Proof. Consider the 3-suns
(toi, 6«, f + 6«; 4 + 6«, to3, to2),
(toi, 1 + 6«, f + 1 + 6«; 5 + 6«, to4, to2),
(toi, 2 + 6«, f +2 + 6«; f + 3 + 6«, to4, to3),
(to2, 1 + 6«, 6«; f + 3 + 6«, to3, to4),
(to2, 2 + 6«, 3 + 6«; f +4 + 6«, 1 + 6«, to4),
(to2, 5 + 6«, 4 + 6«; f +5 + 6«, 6 + 6«, 3 + 6«),
(to3, 3 + 6«, f + 3 + 6«; 2 + 6«, toi, f + 2 + 6«),
(to3, 4 + 6«, f +4 + 6«; f + 6«, to4, toi),
(to3, 5 + 6«, f +5 + 6«; f + 1 + 6«, to4, toi),
(TO4, f + 1 + 6«, f + 2 + 6«; f +3 + 6«, f +6«, TO2),
(TO4, f +4 + 6«, f + 5 + 6«; f +6«, f + 3 + 6«, f + 6 + 6«),
for « = 0,1,..., if - 1.	□
Lemma 2.7. Let u be even, u > 8. Then the graph (Zf U {toi, to2, ..., to6}, {1, f }} can
be decomposed into 3-suns.
Proof. Consider the 3-suns
(toi, 2«, 1 + 2«; f +2 + 2«, f + 2«, TO3), « = 0,1,..., f - 2,
(toi, f - 2, f - 1; f, u - 2, TO3),
(to2, 1 + 2«, f + 1 + 2«; 2«, to6, toi), « = 0,1,..., f - 1, (TO3, f + 1 + 2«, f +2«; 2«, TO6, TOf), « = 0,1,..., f - 1,
(to4, 1 + 2«, 2 + 2«; f + 2 + 2«, to5, to6), « = 0,1,..., f - 1,
(TO5, f + 1 + 2«, f + 2 + 2«; 2 + 2«, TO4, TO6), « = 0,1,..., f - 1.	□
Lemma 2.8. Let u = 0 (mod 12). Then the graph (Zf U {toi, to2, ..., to7}, {1, f}}
can be decomposed into 3-suns.
Proof. Consider the 3-suns
(toi, 6«, f + 6«; 4 + 6«, to7, to2),
(toi, 1 + 6«, f + 1 + 6«; f + 3 + 6«, to7, to4),
(toi, 2 + 6«, f + 2 + 6«; f + 5 + 6«, to5, to2),
(to2, 3 + 6«, f + 3 + 6«; 6«, toi, to4),
(to2, 4 + 6«, f + 4 + 6«; 2 + 6«, to7, toi),
(to2, 5 + 6«, f + 5 + 6«; f + 1 + 6«, toi, to7),
(to3, 6«, 1 + 6«; f + 6«, to5, to6),
(to3, 2 + 6«, 3 + 6«; f + 2 + 6«, to7, to6),
(to3, 4 + 6«, 5 + 6«; f + 5 + 6«, to5, to6),
(to4, 1 + 6«, 2 + 6«; f + 6 + 6«, to2, to6),
(to4, 3 + 6«, 4 + 6«; f + 4 + 6«, to7, to6),
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(to4, 5 + 6«, 6 + 6«; f + 5 + 6«, to7, to6), (tos, f +6«, f + 1 + 6«; 1 + 6«, TO7, tos), (tos, f + 2 + 6«, u + 3 + 6«; 3 + 6«, to7, tos), (tos, f + 4 + 6«, u + 5 + 6«; 5 + 6«, TO7, f + 6 + 6«), (to6, f + 1 + 6«, f + 2 + 6«; f + 5 + 6«, to7, TO4), (to6, f +3 + 6«, f +4 + 6«; f + 6 + 6«, TO7, tos),
for « = 0,1,..., 12 - 1.	□
Lemma 2.9. Let u = 0 (mod 12). Then the graph (Zf U {toi, to2, tos}, {1,2, f}} can be decomposed into 3-suns.
Proof. Consider the 3-suns
(toi, 6«, 1 + 6«; f + 1 + 6«, f + 6«, 3 + 6«),
(toi, 2 + 6«, 3 + 6«; f + 5 + 6«, f +2 + 6«, 5 + 6«),
(toi, 4 + 6«, 5 + 6«; f + 2 + 6«, f +4 + 6«, 7 + 6«),
(toi, f + 3 + 6«, f + 4 + 6«; f + 6«, f +2 + 6«, to2),
(to2, 1 + 6«, f + 1 + 6«; f +3 + 6«, 2 + 6«, f + 2 + 6«),
(to2, 3 + 6«, 4 + 6«; 2 + 6«, f + 3 + 6«, 6 + 6«),
(to2, 5 + 6«, f + 5 + 6«; f +2 + 6«, 6 + 6«, f + 6 + 6«),
(tos, 2 + 6«, 6«; 1 + 6«, 4 + 6«, to2),
(tos, f + 2 + 6«, f + 6«; 4 + 6«, f + 4 + 6«, TO2),
(tos, f + 1 + 6«, f + 3 + 6«; 3 + 6«, f +6«, f + 5 + 6«),
(tos, f + 5 + 6«, f + 4 + 6«; 5 + 6«, f +7 + 6«, f + 6 + 6«),
for « = 0,1,..., 12 - 1.	□
The following lemma "combines" one infinity point with one difference d = f, f such that gcdft d) = 0 (mod 3) (therefore, u = 0 (mod 3)).
Lemma 2.10. Let u = 0 (mod 3) and d G Du \ { f, f} such that p = gcdfM d) = 0 (mod 3). Then the graph (Zf U {to}, {d}} can be decomposed into 3-suns.
Proof. The subgraph (Zf, {d}} can be decomposed into f cycles of length p = 3q, q > 2.
If q > 2, let (xi, x2,..., xSq) be a such cycle and consider the 3-suns
(to, X2+3i, X3+3i; X7+3i, Xi+Si, X4+Si),
for « = 0, 1, . . . , q - 1 (where the sum is modulo 3q).
If q = 2, let (xij),x2j),xSj),x4j),x5j),x6j)), j =0,1,..., f - 1, be the 6-cycles decomposing (Zf, {d}} and consider the 3-suns
(nn x(j) x(j)-x(j + i) x(j) x(j))
, x2 , xs ; xi , xi , x4 ), x(j) x(j)-x(j+i) x(j) x(j))
x^ , x6 ; x4 , x4 , xi ),
for j =0,1,..., f - 1 (where the sums are modulo f).	□
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Subsequent Lemmas 2.11-2.14 allow to decompose (Zu U H, D), where |H| = 1, 2, 3,5, |D| = 6 - |H| and f G D; here, u and D are any with the unique condition that if D contains at least three differences d^ d2, d3, then d3 = d2 — d1 or di + d2 + d3 = u.
Lemma 2.11. Let di, d2, d3, d4, d5 G Du\{ U} such that d3 = d2 — di or di +d2 +d3 = u. Then the graph (Zu U {to}, {di, d2, d3, d4, d5}) can be decomposed into 3-suns.
Proof. If d3 = d2 — di, consider the orbit of
(di, d2,0; to, d2 + d5, d4)
(or (di, d2, 0; to, d2 + d5, —d4), if d2 + d5 = d4) under Zu. If di + d2 + d3 = u, consider the orbit of
(—di, d2, 0; to, d2 + d5, d4)
(or (—di, d2, 0; to, d2 + d5, —d4), if d2 + d5 = d4) under Zu.
□
Lemma 2.12. Let di, d2, d3, d4 G Du \ {u} such that d3 = d2 — di or di + d2 + d3 = u. Then the graph (Zu U {to1, to2}, {di, d2, d3, d4}) can be decomposed into 3-suns.
Proof. Consider the orbit of (d1, d2,0; to1, to2, d4) or (—d1, d2,0; to1, to2, d4) under Zu when, respectively, d3 = d2 — d1 or d1 + d2 + d3 = u.	□
Lemma 2.13. Let d1, d2, d3 G Du \ {|} such that d3 = d2 — d1 or d1 + d2 + d3 = u. Then the graph (Zu U {to1, to2, to3}, {d1, d2, d3}) can be decomposed into 3-suns.
Proof. Consider the orbit of (d1, d2,0; to1, to2, to3) or (—d1, d2,0; to1, to2, to3) under Zu when, respectively, d3 = d2 — d1 or d1 + d2 + d3 = u.	□
Lemma 2.14. Let d G Du \ {}, the graph (Zu U {to1, to2, to3, to4, to5}, {d}) can be decomposed into 3-suns.
Proof. The subgraph (Zu, {d}) is regular of degree 2 and so can be decomposed into l-cycles, l > 3. Let (x1, x2,..., x;) be a such cycle. Put l = 3q + r, with r = 0,1, 2, and consider the 3-suns with the sums modulo l
(toi, xi+3i, X2+3i; X3+3i, TO4, TO5), (TO2, X2+3i, X3+3j; X4+3j, TO4, TO5), (TO3, X3+3j, X4+3j; X5+3j, TO4, TO5),
., q — 2, q > 2, plus the following blocks as the case may be.
for i = 0,1,. If r = 0,
If r = 1,
(TO1, X3q-2, X3q-i; X3q, TO4, TO5), (TO2, X3q_i, X3q; Xi, TO4, TO5), (TO3,X3q,Xi; X2, TO4, TO5) .
(TO1, X3q_2, X3q_i; X3q+1, TO4, TO5), (TO2,X3q_1, X3q; Xi, TO4, TOi), (TO3, X3q, X3q+i; X2 , TO4, TO2), (TO5, X3q+i, Xi; X3q, TO4, TO3).
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If r = 2,
(^1, X3q-2, X3q-i; X3q+2, TO4, TO5), (^2, X3q-1, X3q; X1, TO4, TO5), (^3,X3q,X3q+i; X2, TO1, TO2),
(TO4, X3q+1, X3q+2; X3q, ^1, TO3),
(^5,X3q+2,X1; X3q+1, TO2, TO3).	□
Finally, after settling the infinity points by using the above lemmas, if u is large we need to decompose the subgraph (Zu, L), where L is the set of the differences unused (difference leave). Since by applying Lemmas 2.1 - 2.13 it could be necessary to use the differences 1,
2	or 4, while Lemma 2.14 does not impose any restriction, it is possible to combine infinity points and differences in such a way that the difference leave L is the set of the "small" differences, where 1, 2 or 4 could possibly be avoided.
Lemma 2.15. Let a e {0,4,8} and u, s be positive integers such that u > 12s + a. Then there exists a decomposition of (Zu, L) into 3-suns, where:
i)	a = 0 and L = [1, 6s];
ii)	a = 4 and L = [3, 6s + 2];
iii)	a = 8 and L = [3, 6s + 4] \ {4, 6s + 3}.
Proof.
i)	Consider the orbits (Sj) under Zu, where Sj = (5s + 1 + j, 5s - j, 0; 3s, s, u -2 - 2j), j =0,1,...,s - 1.
ii)	Consider the orbits in i), where (S0) is replaced with the orbit of (6s+1, 4s, 0; s, 9s,
6s + 2).
iii)	Consider the orbits in i), where the orbits (S0) and (S1) are replaced with the orbits
of (6s + 1, 4s, 0; s, 9s, 6s + 4) and (5s + 2, 5s - 1, 0; 3s, s, 6s + 2).	□
3	The main result
Let (X, S) be a 3SS(n) and m = 0,1,4, 9 (mod 12).
Lemma 3.1. If (X, S) is embedded in a 3-sun system of order m > n, then m > | n +1.
Proof. Suppose (X, S) is embedded in (X', S'), with |X'| = m = n + u (u positive integer). Let c be the number of 3-suns of S' each of which contains exactly i edges in X' \ X. Then J2i=1 i x ci = (2) and J25=1(6 - i)cj = u x n, from which it follows 6c2 + 12c3 + 18c4 + 24c5 +30c6 = "(5"-2"-5) andsou > |n +1 andm > |n +1. □
By previous Lemma:
1.	if n = 60k + 5r, r = 0, 5, 8, 9, then m > 84k + 7r +1;
2.	if n = 60k + 5r + 1, r = 0, 3,4, 7, then m > 84k + 7r + 3;
3.	if n = 60k + 5r + 2, r = 2, 7,10,11, then m > 84k + 7r + 4;
4.	if n = 60k + 5r + 3, r = 2, 5, 6, 9, then m > 84k + 7r + 6;
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5. if n = 60k + 5r + 4, r = 0,1,4, 9, then m > 84k + 7r + 7.
In order to prove that the necessary conditions for embedding a 3-sun system (X, S) of order n in a 3-sun system of order m = n + u, u > 0 are also sufficient, the graph (Zu U X, Du) will be expressed as a union of edge-disjoint subgraphs (Zu U X, Du) = (Z„ U X, D) U (Zu, L), where L = D„ \ D is the difference leave, and (Z„ U X, d) (if necessary, expressed itself as a union of subgraphs) will be decomposed by using Lemmas 2.1-2.14, while if L = 0, (Zu, L) will be decomposed by Lemma 2.15. To obtain our main result we will distinguish the five cases 1.-5. listed before by giving a general proof for any k > 0 with the exception of a few cases for k = 0, which will be indicated by a star * and solved in Appendix. Finally, note that:
a)	u = 0,1,4, or 9 (mod 12), if n = 0 (mod 12);
b)	u = 0, 3, 8, or 11 (mod 12), if n = 1 (mod 12);
c)	u = 0, 5, 8, or 9 (mod 12), n = 4 (mod 12);
d)	u = 0, 3,4, or 7 (mod 12), if n = 9 (mod 12).
Proposition 3.2. For any n = 60k + 5r, r = 0, 5, 8,9, there exists a decomposition of Kn+u \ Kn into 3-sunsfor every admissible u > 24k + 2r + 1.
Proof. Let X = {«^ to2,..., «60fc+5r}, r = 0, 5, 8, 9, and u = 24k + 2r + 1 + h, with h > 0. Set h = 12s + l, 0 < l < 11 (l depends on r), and distinguish the following cases.
Case 1: r = 0, 5, 8,9 and l = 0 (odd u).
Write (Zu U X, Du) = (Zu U X, D) U (Zu, L), where D = [6s + 1,12k + r + 6s], |D| = 12k + r, and L = [1,6s], and apply Lemmas 2.14 and 2.15.
Case 2: r = 0, 9 and l = 8 (odd u). Write
(Zu U X, Du) = (Zu U j^i, <^2, «3}, {2, 6s + 3, 6s + 5}) U
(Zu U {<4}, {1}) U (Zu U {<5}, {6s + 4}) U
(Zu U (X \ {«1, «2, «3, «4, «5}), D') U (Zu, L),
where D' = [6s + 6,12k + r + 6s + 4], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.13, 2.10, 2.14 and 2.15.
Case 3: r = 5, 8 and l = 4 (odd u). Write
(Zu U X, Du) = (Zu U {«1, «2, «3, «4}, {2,4}) U (Zu U {«5}, {1}) U
(Zu U (X \ {«1, «2, «3, «4, «5}), D') U (Zu, L),
where D' = [6s + 3,12k + r + 6s + 2] \ {6s + 4}, |D'| = 12k + r - 1, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.3, 2.10, 2.14 and 2.15.
Case 4: r = 0, 8 and l = 3 (even u). Write
(Zu U X, Du) = (Zu U {«1, «2, «3}, {1, u}) U (Zu U {«4, «5}, {2}) U
(Zu U (X \ {«1, «2, «3, «4, «5}), D') U (Zu, L),
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where D' = [6s + 3,12k + r + 6s + 1], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.1, 2.14 and 2.15.
Case 5: r = 0 and l =11 (even u). Write
(Zu u X, Du> = (Zu u {»1, »2, »3}, {1, 2, u}) U
(Zu U {»4, »5}, {4, 6s + 3, 6s + 5, 6s + 7}> U
(Zu U (X \ {»1, »2, »3, »4, »5}), D'> U (Zu, L),
where D' = [6s + 6,12k + 6s + 5] \ {6s + 7}, |D'| = 12k - 1, and L = [3,6s + 4] \ {4, 6s + 3}, and apply Lemmas 2.9, 2.12, 2.14 and 2.15.
Case 6: r = 5 and l = 1 (even u). Write
(Zu U X, Du> = (Zu U {»1, »2, . . . , »6}, {1, u}> U
(Zu U {»7, »8, »9, »10}, {2}> U
(Zu U (X \ {»1, »2,..., »10}), D'> U (Zu, L>,
where D' = [6s + 3,12k + 6s + 5], |D'| = 12k + 3, and L = [3, 6s + 2], and apply Lemmas 2.7, 2.2, 2.14 and 2.15.
Case 7: r = 5,9 and l = 9 (even u). Write
(Zu U X, Du> = (Zu U {»1, »2, »3}, {1, u}> U
(Zu U {»4, »5}, {2, 6s + 3, 6s + 4, 6s + 5}> U
(Zu U (X \ {»1, »2, »3, »4, »5}), D'> U (Zu, L>,
where D' = [6s + 6,12k + r + 6s + 4], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.12, 2.14 and 2.15.
Case 8: r = 8 and l = 7 (even u). Write
(Zu U X, Du> = (Zu U {»1, »2, »3}, {1, 2, u}> U
(Zu U {»4}, {4}> U (Zu U {»5}, {6s + 5}> U
(Zu u (X \ {»1, »2, »3, »4, »5}), D' >U(Zu,L>,
where D' = [6s + 3,12k + 6s + 11] \ {6s + 4, 6s + 5}, |D'| = 12k + 7, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.9, 2.10, 2.14 and 2.15.
Case 9: r = 9 and l = 5 (even u). Write
(Zu U X, Du> = (Zu U {»1, »2, »3}, {1, u}> U (Zu U {»4}, {2}> U
(Zu U {»5}, {4}> U (Zu U (X \ {»1, »2, »3, »4, »5}), D'> U (Zu, L>,
where D' = [6s + 3,12k + 6s + 11] \ {6s + 4}, |D'| = 12k + 8, and L = [3,6s + 4] \ {4, 6s + 3}, and apply Lemmas 2.5, 2.10, 2.14 and 2.15.	□
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Proposition 3.3. For any n = 60k + 5r + 1, r = 0,3,4, 7, there exists a decomposition of \ K„ into 3-sunsfor every admissible u > 24k + 2r + 2.
Proof. Let X = (œj, œ2, ..., œ60k+5r+i}, r = 0, 3, 4, 7, and u = 24k + 2r + 2 + h, with h > 0. Set h = 12s + l, 0 < l < 11, and distinguish the following cases.
Case 1: r = 0, 3 and l = 1 (odd u).
Write (Zu u X, Du) = (Zu U (œ}, (6s + 2}) U (Zu U (X \ (œ}), D') U (Zu, L), where D' = [6s + 1,12k + r + 6s +1] \ (6s + 2}, |D'| = 12k + r, and L = [1,6s], and apply Lemmas 2.10, 2.14 and 2.15.
Case 2: r = 0, 3,4,7 and l = 9 (odd u). Write
(Zu U X, Du) = (Zu U (œi, œ2, œ3}, (1, 6s + 3, 6s + 4}) U (Zu U (œ4, œ5, œ6}, (2, 6s + 5, 6s + 7}) U
(Zu U (X \ (œi, œ2,..., œ6}), D') U (Zu, L),
where D' = [6s + 6,12k + r + 6s + 5]\(6s + 7}, |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.13, 2.14 and 2.15.
Case 3: r = 4*, 7 and l = 5 (odd u). Write
(Zu U X, Du) = (Zu U (œi, œ2, œ3, œ^, (2,4}) U (Zu U (»5}, (1}) U
(Zu U (œ6}, (6s + 8}) U (Zu U (X \ (œi, œ2,..., œ6}), D') U (Zu, L),
where D' = [6s + 3,12k + r + 6s + 3] \ (6s + 4,6s + 8}, |D'| = 12k + r - 1, and L = [3,6s + 4] \ (4, 6s + 3}, and apply Lemmas 2.3, 2.10, 2.14 and 2.15.
Case 4: r = 0,4 and l = 6 (even u). Write
(Zu U X, Du) = (Zu U (œi, œ2, œ3}, (1, f}) U
(Zu U (œ4, œ5, œ6}, (2, 6s + 3, 6s + 5}) U
(Zu U (X \ (œi, œ2,..., œ6|), D') U (Zu, L),
where D' = [6s + 4,12k + r + 6s + 3]\(6s + 5}, |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.13, 2.14 and 2.15.
Case 5: r = 0 and l = 10 (even u). Write
(Zu U X, Du) = (Zu U (œi, œ2,..., œ6}, (1, u}) U
(Zu U (œr, œg, œg, œio}, (2}) U (Zu U (œii}, (4, 6s + 3, 6s + 5, 6s + 6, 6s + 7}) U
(Zu U (X \ (œi, œ2,..., œii}), D') U (Zu, L),
where D' = [6s + 8,12k + 6s + 5], |D'| = 12k - 2, and L = [3,6s + 4] \ (4,6s + 3}, and apply Lemmas 2.7, 2.2, 2.11, 2.14 and 2.15.
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Case 6: r = 3,7 and l = 0 (even u). Write
(Zu u X, Du> = (Zu u {toi, TO2,..., TOe}, {1, u}> U
(Z„ U (X \ {toi, ^2,..., TOe}), D'> U (Zu, L>,
where D' = {2} U [6s + 3,12k + r + 6s], |D'| = 12k + r - 1, and L = [3,6s + 2], and apply Lemmas 2.7, 2.14 and 2.15.
Case 7: r = 3 and l = 4 (even u). Write
(Zu U X, Du > = (Zu U {TOi, TO2, TO3, TO4}, {1, u }> U (Zu U {TO5}, {2}> U
(Zu U {TOe}, {6s + 5}> U (Zu U (X \ {TOi, TO2,..., TOe}), D'> U (Z„, L>,
where D' = [6s + 3,12k + 6s + 5] \ {6s + 5}, |D'| = 12k + 2, and L = [3, 6s + 2], and apply Lemmas 2.6, 2.10, 2.14 and 2.15.
Case 8: r = 4 and l = 2 (even u). Write
(Zu U X, Du > = (Zu U {TOi, TO2, TO3, TO4}, {1, f }> U (Zu U {TO5, TOe}, {2}> U
(Zu U (X \ {TOi, TO2, . . . , TOe}), D'> U (Zu, L>,
where D' = [6s + 3,12k + 6s + 5], |D'| = 12k + 3, and L = [3, 6s + 2], and apply Lemmas 2.6, 2.1, 2.14 and 2.15.
Case 9: r = 7 and l = 8 (even u). Write
(Zu U X, Du> = (Zu U {TOi, TO2, TO3}, {1, 2, u}> U
(Zu U {TO4, TO5, TOe}, {4, 6s + 3, 6s + 7}> U
(Zu U (X \ {TOi, TO2,..., TOe}), D'> U (Zu, L>,
where D' = [6s + 5,12k + 6s + 11] \ {6s + 7}, |D'| = 12k + 6, and L = [3,6s + 4] \ {4, 6s + 3}, and apply Lemmas 2.9, 2.13, 2.14 and 2.15.	□
Proposition 3.4. For any n = 60k + 5r + 2, r = 2, 7,10,11, there exists a decomposition of Kn+u \ Kn into 3-sunsfor every admissible u > 24k + 2r + 2.
Proof. Let X = {to1, to2,..., TOe0k+5r+2}, r = 2, 7,10,11, and u = 24k + 2r + 2 + h, with h > 0. Set h = 12s + l, 0 < l < 11, and distinguish the following cases.
Case 1: r = 2,11 and l = 3 (odd u). Write
(Zu U X, Du> = (Zu U {TOi}, {6s + 2}> U (Zu U {TO2}, {6s + 4}> U
(Zu U (X \{TOi, TO2}),D'>U(Zu,L>,
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where D' = [6s + 1,12k + r + 6s + 2] \{6s + 2,6s + 4}, |D'| = 12k + r, and L = [1, 6s], and apply Lemmas 2.10, 2.14 and 2.15.
Case 2: r = 2, 7,10,11 and l = 7 (odd u). Write
(Z„ U X, D„) = (Z„ U {wb ^2}, {1, 2, 6s + 3, 6s + 4}) U
(Z„ U (X \ {^1, ^2}), D') U (Z„, L),
where D' = [6s + 5,12k + r + 6s + 4], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.12, 2.14 and 2.15.
Case 3: r = 7,10 and l =11 (odd u). Write
(Z„ U X, D„) = (Z„ U {wb TO2, }, {1, 6s + 3, 6s + 4}) U
(Z„ U {^4, ^5, we}, {2, 6s + 5, 6s + 7}) U (Z„ U {wr}, {6s + 8}) U
(Z„ U (X \ {^1, ^2,..., tot}), D') U (Z„, L),
where D' = [6s + 6,12k + r + 6s + 6] \ {6s + 7,6s + 8}, |D'| = 12k + r - 1, and L = [3,6s + 2], and apply Lemmas 2.13, 2.10, 2.14 and 2.15.
Case 4: r = 2 and l = 6 (even u). Write
(Z„ U X, D„) = (Z„ U {^1, W2, W3, W4}, {1, u}) U
(Z„ U {^5, we, Tot}, {2, 6s + 3, 6s + 5}) U
(Z„ U (X \ {W1, W2,..., wr}), D') U (Z„, L),
where D' = [6s + 4,12k + 6s + 5] \ {6s + 5}, |D'| = 12k + 1, and L = [3, 6s + 2], and apply Lemmas 2.6, 2.13, 2.14 and 2.15.
Case 5: r = 2,10 and l = 10 (even u). Write
(Z„ U X, D„) = (Z„ U {w 1, W2,..., we}, {1, u}) U
(Z„ U {wr}, {2, 6s + 3, 6s + 4, 6s + 5, 6s + 6}) U
(Z„ U (X \ {w1, w2,..., wr}), D') U (Z„, L),
where D' = [6s + 7,12k + r + 6s + 5], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.7, 2.11, 2.14 and 2.15.
Case 6: r = 7,11 and l = 4 (even u). Write
(Z„ U X, D„) = (Z„ U {w 1, w2, w3}, {1, u}) U
(Z„ U {w4, w5, we, wr}, {2,4}) U
(Z„ U (X \ {w1, w2,..., wr}), D') U (Z„, L),
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where D' = [6s + 3,12k + r + 6s + 2] \ {6s + 4}, |D'| = 12k + r - 1, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.5, 2.3, 2.14 and 2.15.
Case 7: r = 7 and l = 8 (even u). Write
(Zu u X, Du) = (Zu u {»1, »2, »3}, {1, u}> u
(Zu u {»4, »5, »6}, {2, 6s + 3, 6s + 5}) u (Zu U {»7}, {6s + 7}) U
(Zu u (X \ {»1, »2,..., »7}), D') u (Zu, L),
where D' = [6s + 4,12k + 6s + 11] \{6s + 5,6s + 7}, |D'| = 12k + 6, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.13, 2.10, 2.14 and 2.15.
Case 8: r =10 and l = 2 (even u). Write
(Zu u X, Du) = (Zu u {»1, »2,..., »6}, {1, u}) u (Zu u {»7}, {2}) u
(Zu u (X \ {»1, »2,..., »7}), D') u (Zu, L),
where D' = [6s + 3,12k + 6s + 11], |D'| = 12k + 9, and L = [3, 6s + 2], and apply Lemmas 2.7, 2.10, 2.14 and 2.15.
Case 9: r =11 and l = 0 (even u). Write
(Zu u X, Du) = (Zu u {»1, »2,..., »7}, {1, u}) u
(Zu u (X \ {»1, »2,..., »7}), D') u (Zu, L),
where D' = {2} u [6s + 3,12k + 6s + 11], |D'| = 12k + 10, and L = [3,6s + 2], and apply Lemmas 2.8, 2.14 and 2.15.	□
Proposition 3.5. For any n = 60k + 5r + 3, r = 2,5,6, 9, there exists a decomposition of Kn+u \ Kn into 3-sunsfor every admissible u > 24k + 2r + 3.
Proof. Let X = {»1, »2,..., »60fc+5r+3}, r = 2, 5, 6, 9, and u = 24k + 2r + 3 + h, with h > 0. Set h = 12s + l, 0 < l < 11, and distinguish the following cases.
Case 1: r = 2,5, 6, 9 and l = 4 (odd u). Write
(Zu u X, Du) = (Zu u {»1, »2, »3}, {1, 6s + 3, 6s + 4}) u
(Zu u (X \ {»1, »2, »3})
where D' = {2} u [6s + 5,12k + r + 6s + 3], |D'| = 12k + r, and L apply Lemmas 2.13, 2.14 and 2.15.
Case 2: r = 2,5 and l = 8 (odd u). Write
(Zu u X, Du) = (Zu u {»1, »2}, {1, 6s + 3, 6s + 4, 6s + 5}) u
(Zu u {»3}, {2}) u (Zu u (X \ {»1, »2, »3}), D') u (Zu, L),
, D' )u(Zu,L), = [3,6s + 2], and
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where D' = [6s + 6,12k + r + 6s + 5], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.12, 2.10, 2.14 and 2.15.
Case 3: r = 6, 9 and l = 0 (odd u). If s = 0, then write
(Z„ U X, D„} = (Z„ U {TO1, TO2, . . . , TOg}, {1, u}} U
(Z„ U (X \{TO1, TO2,..., TOg}),D'},
where D' = [2,12k + r +1] \ {u}, |D'| = 12k + r - 1, and apply Lemmas 2.4 and 2.14. If s > 0, then write
(Z„ U X, D„} = (Z„ U {TO1, TO2, TO3}, {1, 5s, 5s + 1}} U
(Z„ U {TO4, TO5, Tog}, {2, 6s + 1, 6s + 3}} U (Z„ U {to7}, {6s + 2}} U
(Z„ U {TOg}, {6s + 4}} U (Z„ U (X \ {toi, TO2,..., TOg}), D'} U (Z„, L},
where D' = {2s + 1,4s} U [6s + 5,12k + r + 6s + 1], |D'| = 12k + r - 1, and L = [3, 6s] \ {2s + 1,4s, 5s, 5s + 1}, and apply Lemmas 2.13, 2.10 and 2.14 to decompose the first five subgraphs, while to decompose the last one apply Lemma 2.15 i) and delete the orbit (So).
Case 4: r = 2, 6 and l = 1 (even u). Write
(Z„ U X, D„} = (Z„ U {TO1, TO2, TO3}, {1, u}} U
(Z„ U (X \ {TO1, TO2, TO3}), D'} U (Z„, L},
where D' = {2} U [6s + 3,12k + r + 6s + 1], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.5, 2.14 and 2.15.
Case 5: r = 2* and l = 5 (even u). Write
(Z„ U X, D„} = (Z„ U {TO1, TO2,..., TOG}, {1, u}} U
(Z„ U {TO7, TOg, TO9, TO10}, {2}} U (Z„ U {to 11, to 12, TO13}, {4, 6s + 3, 6s + 7}} U
(Z„ U (X \ {TO1, TO2, ..., TO13}), D'} U (Z„, L},
whereD' = [6s + 5,12k + 6s + 5]\{6s + 7}, |D'| = 12k, andL = [3, 6s + 4]\{4, 6s + 3}, and apply Lemmas 2.7, 2.2, 2.13, 2.14 and 2.15.
Case 6: r = 5, 9 and l = 7 (even u). Write
(Z„ U X, D„} = (Z„ U {TO1, TO2,..., TOG}, {1, u}} U
(Z„ U {to7, TOg}, {2, 6s + 3, 6s + 4, 6s + 5}} U
(Z„ U (X \ {to1, TO2,..., TOg}), D'} U (Z„, L},
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where D' = [6s + 6,12k + r + 6s + 4], |D'| = 12k + r - 1, and L = [3, 6s + 2], and apply Lemmas 2.7, 2.12, 2.14 and 2.15.
Case 7: r = 5 and l =11 (even u). Write
(Z„ u X, D„) = (Z„ u {»1, »2, »3, »4}, {1, u}) u
(Z„ U {»5, »6}, {2, 6s + 3, 6s + 5, 6s + 6}) U (Z„ U {»7}, {4}) U (Z„ U {»g}, {6s + 7}) U (Z„ U (X \ {»1, »2,..., »8}), D') U (Z„, L),
where D' = [6s + 8,12k + 6s + 11], |D'| = 12k + 4, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.6, 2.12, 2.10, 2.14 and 2.15.
Case 8: r = 6 and l = 9 (even u). Write
(Z„ U X, D„) = (Z„ U {»1, »2, »3}, {1, u}) U
(Z„ U {»4, »5, »6}, {2, 6s + 3, 6s + 5}) U (Z„ U {»7}, {4}) U
(Z„ U {»g}, {6s + 7}) U (Z„ U (X \ {»1, »2,..., »g}), D') U (Z„, L),
where D' = [6s + 6,12k + 6s + 11] \ {6s + 7}, |D'| = 12k + 5, and L = [3,6s + 4] \ {4, 6s + 3}, and apply Lemmas 2.5, 2.13, 2.10, 2.14 and 2.15.
Case 9: r = 9 and l = 3 (even u). Write
(Z„ U X, D„) = (Z„ U {»1, »2, »3}, {1, 2, u}) U
(Z„ U (X \ {»1, »2, »3}), D') U (Z„, L),
where D' = [6s + 3,12k + 6s + 11], |D'| = 12k + 9, and L = [3, 6s + 2], and apply Lemmas 2.9, 2.14 and 2.15.	□
Proposition 3.6. For any n = 60k + 5r + 4, r = 0,1,4, 9, there exists a decomposition of Kn+„ \ Kn into 3-sunsfor every admissible u > 24k + 2r + 3.
Proof. Let X = (œj, œ2,..., ^60fe+5r+4}, r = 0,1,4, 9, and u = 24k + 2r + 3 + h, with h > 0. Set h = 12s + l, 0 < l < 11, and distinguish the following cases.
Case 1: r = 0,1*, 4, 9 and l = 2 (odd u). Write
(Z„ U X, D„) = (Z„ U{»1, »2, »3, »4}, {2,4})U
(Z„ U (X \ {»1, »2, »3, »4}), D') U (Z„, L),
where D' = {1, 6s + 3} U [6s + 5,12k + r + 6s + 2], |D'| = 12k + r, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.3, 2.14 and 2.15.
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Case 2: r = 0, 9 and l = 6 (odd u). Write
(Zu U X, Du) = (Zu U {»i, »2, »3}, {1, 6s + 3, 6s + 4}) U
(Z„ U {»4}, {2}) U (Zu U (X \ {»i, »2, »3, »4}), D') U (Zu, L),
where D' = [6s + 5,12k + r + 6s + 4], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.13, 2.10, 2.14 and 2.15.
Case 3: r = 1,4 and l = 10 (odd u). Write
(Zu U X, Du) = (Zu U {»1, »2}, {1, 6s + 3, 6s + 5, 6s + 6}) U (Z„ U {»3}, {2}) U (Z„ U {»4}, {6s + 4}) U
(Z„ U (X \ {»1, »2, »3, »4}), D') U (Zu, L),
where D' = [6s + 7,12k + r + 6s + 6], |D'| = 12k + r, and L = [3,6s + 2], and apply Lemmas 2.12, 2.10, 2.14 and 2.15.
Case 4: r = 0,4 and l = 5 (even u). Write
(Zu U X, Du} = (Zu U {toi, TO2, . . . , TO6}, {1, U}} U
(Zu U {to7, to8, to9}, {2, 6s + 3, 6s + 5}} U
(Zu U (X \ {TO!, TO2, . . . , TO9}), D'} U (Zu, L},
whereD' = [6s + 4,12k + r + 6s + 3]\{6s + 5}, |D'| = 12k + r- 1, andL = [3, 6s + 2], and apply Lemmas 2.7, 2.13, 2.14 and 2.15.
Case 5: r = 0 and l = 9 (even u). Write
(Zu U X, Du} = (Zu U {TO1, TO2, TO3, TO4}, {1, u }} U
(Zu U {TO5, TO6, TO7}, {2, 6s + 3, 6s + 5}} U (Zu U {to8}, {4}} U
(Zu U {TO9}, {6s + 7}} U (Zu U (X \ {toi, TO2,..., toq}), D'} U (Zu, L},
whereD' = [6s+6,12k+6s+5]\{6s+7}, |D'| = 12k-1,andL = [3,6s+4]\{4,6s+3}, and apply Lemmas 2.6 , 2.13, 2.10, 2.14 and 2.15.
Case 6: r = 1 and l = 7 (even u). Write
(Zu U X, Du} = (Zu U {TO1, TO2, . . . , TO7}, {1, u}} U
(Zu U {to8, Toq}, {2,4, 6s + 3, 6s + 5}} U
(Zu U (X \ {TO1, TO2, . . . , Toq}), D'} U (Zu, L},
where D' = [6s + 6,12k + 6s + 5], |D'| = 12k, and L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.8, 2.12, 2.14 and 2.15.
G. Lo Faro and A. Tripodi: The Doyen-Wilson theorem for 3-sun systems
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Case 7: r =1,9 and l = 11 (even u). Write
(Z„ U X, Du> = (Z„ U {toi, TO2, tos}, {1, u}> u
(Z„ U {to4}, {2, 4, 6s + 3, 6s + 5, 6s + 6}> U
(Z„ U (X \ {toi, TO2, tos, TO4}), D'> U (Z„, L>,
where D' = [6s + 7,12k + r + 6s + 6], |D'| = 12k + r, and L = [3, 6s + 4] \{4,6s + 3}, and apply Lemmas 2.5, 2.11, 2.14 and 2.15.
Case 8: r = 4 and l = 1 (even u). Write
(Z„ U X, D„> = (Z„ U {toi, TO2, tos, TO4}, {1, u}> U
(Z„ U (X \ {toi, TO2, tos, TO4}), D'> U (Z„, L>,
where D' = {2} U [6s + 3,12k + 6s + 5], |D'| = 12k + 4, and L = [3,6s + 2], and apply Lemmas 2.6, 2.14 and 2.15.
Case 9: r = 9 and l = 3 (even u). Write
(Z„ U X, D„> = (Z„ U {toi, TO2, tos}, {1, u}> U (Z„ U {TO4}, {2}> U
(Z„ U (X \ {toi, TO2, tos, TO4}), D'> U (Z„, L>,
where D' = [6s + 3,12k + 6s + 11], |D'| = 12k + 9, and L = [3, 6s + 2], and apply Lemmas 2.5, 2.10, 2.14 and 2.15.	□
Combining Lemma 3.1 and Propositions 3.2 - 3.6 gives our main theorem.
Theorem 3.7. Any 3SS(n) can be embedded in a 3SS(m) if and only if
m > 7n +1 or m = n.
References
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Appendix
•	n = 21, u = 12s +15 Write
(Zu u X, D„) = (Zu u {TO1, TO2, TOS, TO4}, {2,4}) u
(Zu U {^5}, {1}) u (Zu U {TOe}, {6s + 7}) U (Zu U (X \ {toi, TO2,..., TOe}), {6s + 3, 6s + 5, 6s + 6}) U (Z„, L),
where L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.3, 2.10, 2.14 and 2.15.
•	n =13, u = 12s +12 Write
(Zu U X, Du) = (Zu U {toi, to2, ..., TOe}, {1,6s + 6}) U (Zu U {TO7, TOg, TOq, TO10}, {2}) U (Zu U {TO11, TO12, to 13}, {4, 6s + 3, 6s + 5}) U (Zu, L),
where L = [3, 6s + 4] \ {4,6s + 3}, and apply Lemmas 2.7, 2.2, 2.13 and 2.15.
•	n = 9, u = 12s + 7 Write
(Zu U X, Du) = (Zu U {TO1, TO2, tos, TO4}, {2,4}) U
(Zu U {TO5, TOe, TO7, TOg, TOq}, {1}) U (Zu, L),
where L = [3,6s + 3] \ {4}, and apply Lemmas 2.3, 2.14 and decompose (Zu, L) as in Lemma 2.15 iii), taking in account that |6s + 4|12s+7 = 6s + 3.