圖-設計的研究(III)

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圖-設計計畫結案報告

傅恆霖

2008,8,8

中文摘要：

圖-設計(Graph Design)主要是研究如何把一個圖 H 分割成很多個

彼此互相同構的子圖 G，通常我們用 G|H 表示。由於

Kk|

λ

Kv

對應於

一個

2 ( , , )− v k

λ

設計，

K_k|

λ

K_{m n}_{( )}

對應於 Group Divisible Design (可分組

設計)

GD n m k[ , ; , ]

λ

，所以設計理論的研究自然在圖設計的研究中扮演

非常重要的角色。

以上所提的兩種設計已經廣泛地應用在實驗設計中，所以，我們

的研究除了增加更近代科技領域上的用途之外，也尋求不同的設計，

例如 Grid-Block Design，它在 DNA Library Screening 方面有很重要的

應用；另外，除了設計，也探討 packing 的應用，在同步光學網路及

群試理論的平行式演算法上我們也得到很好的應用。

關鍵字：圖設計、方格圖設計、DNA 排序、同步光學網路、平行群

試演算法。

英文摘要：

The main focus of the study of graph-design is to decompose a graph

H into isomorphic copies of subgraphs G, denoted by G|H. It is

well-known that

K_k|

λ

K_v

is equivalent to the existence of a

2 ( , , )− v k

λ

design and

Kk |

λ

Km n( )

is equivalent to the existence of a group divisible

design

GD n m k[ , ; , ]

λ

. Therefore, the study of combinatorial designs plays

an important role in our study.

The use of combinatorial designs in experiment designs has been

known for many occasions. Thus, we intend to add more which are

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of designs such as grid-block design. Note that this design has its

application on DNA Library Screening (related to DNA sequencing).

Besides, we also utilize the packing of graph to obtain well-constructed

SONET and better disjunct matrices which are the main objectives in

nonadaptive algorithms of group testing.

Keywords：Graph design, Grid-block design, DNA-sequencing, SONET,

Non-adaptive algorithms.

報告內容：

在過去三年中，我們在圖設計的建構方面獲得不少成果，參見附

錄。基本上我們的工作可以分成理論的建構與群試及網路的應用。前

者，除了圖的圈分割[2,3,4,5,7,9,10]之外，我們也完成用配對[6,11]或

較短路徑[8]來覆蓋一個圖。在應用方面，主要的工作之ㄧ是群試理

論中建構 Non-adaptive algorithm 的 disjunct 矩陣，我們適當地利用圖

設計的結果來完成工作[1,12]；另外，利用不同子圖來裝填完全圖也

在同步光學網路上找到很好的應用[13]。

附錄：

1.

A novel use of t-packings in constructing d-disjunct matrices (with F. K. Hwang), Discrete Applied Math., Vol. 154, Issue 12, 2006, 1759-1762.

2.

Maximum Cyclic 4-cycle packings of the complete multipartite graph (with S. L. Wu), J. Combin. Optimization, Number 2-3, Vol. 14(2007), 365-382.

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Rodger), Discrete Math., 308(2008), 2822-2829.

4.

All graphs with maximum degree three whose complements have 4-cycle decompositions (with C. M. Fu, C. A. Rodger and Todd Smith), Discrete Math. 308(2008), 2901-2909.

5.

Maximal sets of Hamitonian cycles in

n

D

(with Liqun Pu and Hao Shen),

Discrete Math., 308(2008), 3706-3710.

6.

On the minimum sets of 1-factors covering a complete multipartite graph (with D. Cariolaro), J. Graph Theory, Vol. 58, Issue 3(2008), 239-250.

7.

The Hamilton-Waterloo problem for two even cycle factors (with Kuo-Cjing Huang), Taiwanese J. Math., Vol. 12, No. 4, 2008, 933-940.

8.

The linear 3-arboricity of K_n_,_n and K (with K. C. Huang and C. H. Yen), _n Discrete Math., 308, Issue 17(2008), 3761-3769.

9.

Multicolored parallelism of Hamiltonian cycles (with Y. H. Lo), Discrete Math., to appear.

10.

C3

JJK

-decompositions of D with quadratic leaves (with Liqun Pu and H. Sheu), _t Discrete Math., to appear.

11.

Covering graphs with matchings of fixed size (with D. Cariolaro), Discrete Math., to appear.

12.

A new construction of 3-separable matrices via improved decoding of Macula construction, Discrete Optim., to appear.

13.

Minimizing SONET ADMs in indirectional WDM rings with grooming rate 7 (with C. Colbourn, G. Ge, A. Ling and Hui-Chuan Lu), SIAM J. Discrete Math., to appear.

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Minimizing SONET ADMs in Unidirectional WDM

Rings with Grooming Ratio 7

Charles J. Colbourn

∗

Hung-Lin Fu

†

Gennian Ge

‡

Alan C.H. Ling

§

Hui-Chuan Lu

¶

November 18, 2007

Abstract

In order to reduce the number of add-drop multiplexers (ADMs) in SONET/WDM net-works using wavelength add-drop multiplexing, certain graph decompositions can be used to form a ‘grooming’ that specifies the assignment of traffic to wavelengths. When traffic among nodes is all-to-all and uniform, the drop cost of such a decomposition is the sum, over all graphs in the decomposition, of the number of vertices of nonzero degree in the graph. The number of ADMs required is this drop cost. The existence of such decompositions with min-imum cost, when every pair of sites employs no more than 1₇ of the wavelength capacity, is determined within an additive constant. Indeed when the number n of sites satisfies n ≡ 1 (mod 3) and n 6= 19, the determination is exact; when n ≡ 0 (mod 3), n 6≡ 18 (mod 24), and n is large enough, the determination is also exact; and when n ≡ 2 (mod 3) and n is large enough, the gap between the cost of the best construction and the cost of the lower bound is independent of n and does not exceed 4.

1 Introduction

Traffic grooming in optical (SONET) rings arises from amalgamating C low rate signals onto a higher capacity wavelength [15, 25, 26]; C is the grooming ratio. Nodes initiate or terminate traffic on a wavelength using an add-drop multiplexer (ADM). Finding the minimum number of add-drop multiplexers (ADMs), A(C, n), required in an n-node SONET ring with grooming ratio

∗_{Computer Science and Engineering, Arizona State University, PO Box 878809, Tempe, AZ 85287-8809, U.S.A.} colbourn@asu.edu

†_{Department of Applied Mathematics, National Chaio Tung University, Hsin Chu, Taiwan, R. O. C.} hlfu@math.nctu.edu.tw

‡_Department _of _Mathematics, _Zhejiang _University, _Hangzhou _310027, _Zhejiang, _P. _R. _China gnge@zju.edu.cn

§_{Computer Science, University of Vermont, Burlington, VT 05405, U.S.A. aling@cems.uvm.edu}

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C, is equivalent to the following problem in graphs [4]: Given a number of nodes n and a grooming ratio C find a partition of the edges of Kninto subgraphs B`, ` = 1, . . . , s with |E(B`)| ≤ C such

thatP

1≤`≤s|V (B`)| is minimum.

Optimal constructions for given grooming ratio C have been obtained using tools of graph and design theory [9]. Results are known for grooming ratio C = 3 [1], C = 4 [5, 23], C = 5

[3], C = 6 [2], C ≤ 1₆n(n − 1) [5], and for large values of C [5]. Related problems have

been studied for variable traffic requirements [8, 14, 22, 27, 29], for fixed traffic requirements [1, 3, 4, 5, 15, 21, 23, 24, 25, 28, 30], and in the case of bidirectional rings [10, 13]. The explicit correspondence between grooming and graph decomposition is developed in detail in [1, 11].

In this paper we consider grooming with grooming ratio 7. In Section 2 we employ linear programming duality to establish a general lower bound on A(7, n). In Section 4 we determine A(7, n) with the possible exception of n = 19 when n ≡ 1 (mod 3). When n ≡ 0 (mod 3) (Section 5) we determine A(7, n) with finitely many possible exceptions except when n ≡ 18 (mod 24); in the latter case we establish a construction whose cost exceeds the lower bound by 1. When n ≡ 2 (mod 3) (Section 6) we develop a set of constructions to establish that, with finitely many possible exceptions, the cost does not exceed the lower bound by more than 4, independent of n.

It is natural to ask why the case when C = 7 is of independent interest. Unlike all cases when C ≤ 6, the graph with the lowest ratio of number of vertices to number of edges does not have C edges; rather it is K4, a 6-edge graph. This necessitates consideration of decompositions

that do not use the minimum number of graphs, and hence determining the minimum number of wavelengths required is quite different than determining the minimum drop cost.

2 The Lower Bounds

We adapt a strategy using linear programming from [12] that was used in [11] to determine both the cost and the structure of certain optimal groomings. A grooming with ratio 7 is a decomposition of Kninto subgraphs each having at most 7 edges. Its drop cost, or just cost, is the sum of the numbers

of vertices of nonzero degree over all graphs in the decomposition. A(7, n) is the minimum drop cost of a grooming of Kn with grooming ratio 7. Figure 1 displays all connected graphs having at

most 7 edges. The naming convention is as follows. For each number q of edges and p of vertices, suppose that there are γq,p nonisomorphic graphs. These are named G`,q,pfor 1 ≤ ` ≤ γq,p.

In a decomposition, let α`,q,pbe the number of occurrences of G`,q,pand let αq,p =P γq,p `=1α`,q,p.

Then because every edge appears in exactly one of the chosen subgraphs,

7 X q=1 8 X p=1 γq,p X `=1 q · α`,q,p= n 2 (1)

In order to minimize drop cost, we must compute

min 7 X q=1 8 X p=1 γq,p X `=1 p · α`,q,p (2)

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Graph G_l_{, ,}_{q p} deg. seq. !_l_{, ,}_{q p} "l, ,q p Graph Gl, ,q p deg. seq. !l, ,q p "l, ,q p 4,6,5 G 33321 1.5 1.5 1,7,5 G 2,7,5 G 3,7,5 G 4,7,5 G 44222 43322 43331 33332 4 2.5 1 1 3.5 2 2 0.5 1,6,6 G 2,6,6 G 3,6,6 G 4,6,6 G 5,6,6 G 6,6,6 G 7,6,6 G 522111 422211 432111 222222 322221 332211 333111 4.5 4.5 3 6 4.5 3 1.5 4.5 4.5 4.5 3 3 3 3 1,7,6 G 2,7,6 G 3,7,6 G 4,7,6 G 5,7,6 G 6,7,6 G 7,7,6 G 442211 522221 422222 532211 432221 433211 332222 4 5.5 5.5 4 4 2.5 4 5 3.5 3.5 3.5 3.5 3.5 2 1,6,7 G 2,6,7 G 3,6,7 G 4,6,7 G 5,6,7 G 6,6,7 G 7,6,7 G 5211111 4221111 4311111 6111111 2222211 3222111 3321111 4.5 4.5 3 3 6 4.5 3 6 6 6 6 4.5 4.5 4.5 1,5,4 G 3322 2 1 1,5,5 G 2,5,5 G 3,5,5 G 4,5,5 G 42211 22222 32221 33211 3.5 5 3.5 2 4 2.5 2.5 2.5 1,7,7 G 2,7,7 G 3,7,7 G 4,7,7 G 5,7,7 G 6,7,7 G 7,7,7 G 8,7,7 G 9,7,7 G 10,7,7 G 4421111 5222111 4222211 5321111 4322111 4331111 2222222 3222221 3322211 3332111 4 5.5 5.5 4 4 2.5 7 5.5 4 2.5 6.5 5 5 5 5 5 3.5 3.5 3.5 3.5 1,5,6 G 2,5,6 G 3,5,6 G 4,5,6 G 5,5,6 G 421111 511111 322111 331111 222211 3.5 3.5 3.5 2 5 5.5 5.5 4 4 3 1,4,4 G 2,4,4 G 2222 3221 4 2.5 2 2 1,4,5 G 2,4,5 G 3,4,5 G 41111 22211 32111 2.5 4 2.5 5 3.5 3.5 1,3,3 G 222 3 1.5 1,7,8 G 2,7,8 G 3,7,8 G 4,7,8 G 5,7,8 G 6,7,8 G 7,7,8 G 8,7,8 G 9,7,8 G 10,7,8 G 11,7,8 G 71111111 44111111 52211111 42221111 62111111 53111111 43211111 22222211 32222111 33221111 33311111 4 4 5.5 5.5 4 4 4 7 5.5 4 2.5 8 8 6.5 6.5 6.5 6.5 6.5 5 5 5 5 1,3,4 G 3333 0 0 1,6,4 G G2,3,4 2211 3111 3 1.5 3 3 1,2,3 G 211 2 2.5 1,6,5 G 2,6,5 G 3,6,5 G 42222 43221 33222 4.5 3 3 3 3 1.5 1,1,2 G 11 1 2

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Figure 1 does not list disconnected graphs, but the cost of a disconnected graph is the sum of the costs of its components, so all feasible decompositions are accounted for. For every graph G`,q,p, we find that p_q ≥ 2₃. Subtract 2₃×(1) from (2) to restate the minimum drop cost A(7, n) as

n(n − 1) 3 + min 7 X q=1 8 X p=1 γq,p X `=1 (p − 2 3q) · α`,q,p (3)

In (3) the triple summation is always nonnegative; it can be zero only when all graphs are isomorphic to K4. However, structural restrictions can prohibit such a selection. In particular,

considering the number n₂ of edges modulo 6,

7 X q=1 8 X p=1 γq,p X `=1 (q mod 6) · α`,q,p ≡        0 (mod 6) if n ≡ 0, 1, 4, 9 (mod 12) 1 (mod 6) if n ≡ 2, 11 (mod 12) 3 (mod 6) if n ≡ 3, 6, 7, 10 (mod 12) 4 (mod 6) if n ≡ 5, 8 (mod 12) (4)

We can relax this congruence to linear inequalities. For example, if n ≡ 3, 6, 7, 10 (mod 12),

8 X p=1   γ3,p X `=1 α`,3,p+ 1 3( X q∈{1,4,7} γq,p X `=1 α`,q,p) + 2 3( X q∈{2,5} γq,p X `=1 α`,q,p)  ≥ 1 (5)

because if there is no graph on three edges, there must be at least three graphs having 1 (mod 3) edges, or one having 1 (mod 3) edges and one having 2 (mod 3) edges.

Every vertex of Knhas degree congruent to n − 1 mod 3; placing a K4 in the decomposition

does not change this congruence class at any vertex, and hence subgraphs other than K4 may be

needed to accommodate these vertex degrees. Let ω`,q,pbe the number of vertices whose degree is

congruent to 1 modulo 3 in G`,q,p, and let τ`,q,pbe the number of vertices whose degree is congruent

to 2 modulo 3. Now if n ≡ 0 (mod 3), every vertex has degree 2 modulo 3, and hence at every vertex there must either be a graph itself having degree 2 modulo 3, or two graphs each having degree 1 modulo 3 (there may be more). And if n ≡ 2 (mod 3), every vertex has degree 1 modulo 3, and hence at every vertex there must either be a graph itself having degree 1 modulo 3, or two graphs each having degree 2 modulo 3. For convenience we write φ`,q,p = 1₂ω`,q,p + τ`,q,p and

ψ`,q,p = ω`,q,p+1₂τ`,q,p. These are tabulated for each graph in Figure 1. We conclude that

P7 q=1 P8 p=1 Pγq,p `=1φ`,q,p· α`,q,p≥ n if n ≡ 0 (mod 3) P7 q=1 P8 p=1 Pγq,p `=1ψ`,q,p· α`,q,p≥ n if n ≡ 2 (mod 3) (6) Theorem 2.1 The cost of an optimal grooming of Knwith grooming ratio 7,A(7, n), is at least

2 3 n 2 if n ≡ 1, 4 (mod 12) 2 3 n 2 + 1 if n ≡ 7, 10 (mod 12) 2 3 n 2 + d n 12e if n ≡ 0, 3, 6, 15, 18, 21 (mod 24) 2 3 n 2 + d n 12e + 1 if n ≡ 9, 12 (mod 24) d2 3 n 2 + 2n 21e if n ≡ 5, 8, 17 (mod 21) orn ≡ 2, 23, 32, 53, 56, 77, 62, 83 (mod 84) d2 3 n 2 + 2n 21e + 1 if n ≡ 14, 35, 20, 41, 44, 65, 74, 11 (mod 84)

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Proof: We follow the strategy in [12]. Form a linear program whose variables are the {α`,q,p}s. min P7 q=1 P8 p=1 Pγq,p `=1(p − 2 3q) · α`,q,p

subject to (4) suitably relaxed, (6), and nonnegativity of each variable (7)

If z? is the minimum, the cost of any grooming must be at least d2₃ n₂ + z?e, since the cost is integral. By forming the dual of (7), any feasible solution to the dual gives a lower bound on all primal feasible solutions, and hence a lower bound on z?.

Case 1: n ≡ 1 (mod 3): When n ≡ 1, 4 (mod 12), the linear program is constrained only by nonnegativity, and the dual optimum is 0. When n ≡ 7, 10 (mod 12), (5) holds. Call its dual variable y1. An assignment y1?is dual feasible if y1? ≤ p − 2 for every graph G`,3,p; y1? ≤ 32(p −

2 3q)

for every graph G`,q,pwith q ∈ {2, 5}; and y?1 ≤ 3(p − 2

3q) for every graph G`,q,pwith q ∈ {1, 4, 7}.

By considering the graphs in Figure 1 the dual optimum of 1 occurs when y?

1 = 1. This raises the

lower bound by 1.

Case 2: n ≡ 0 (mod 3): Consider the inequality from (6), and let y2 be its dual variable. Each

graph G`,q,p leads to the dual inequality φ`,q,py2 ≤ p − 2₃q. The dual optimum of ₁₂n arises when

y?₂ = ₁₂1; the only graph whose dual inequality is binding is G1,7,5with φ1,7,5 = 4 and 5 − 2₃7 = 1₃.

We can compute the slackness of each variable; for α`,q,p, the slackness is p − 2₃q − ₁₂1φ`,q,p. A

unit increase in the variable α`,q,pincreases the dual objective function value by the slackness. The

only variables with slackness at most 1₂ are α2,7,5with slackness 1₈, α3,7,5 and α4,7,5 with slackness 1

4, and α1,5,4 with slackness 1

2. Hence any decomposition of cost less than n 12 +

1

2 consists solely

of graphs in {G`,7,5}. To satisfy (6), α7,5 ≥ dn₄e. If α7,5 ≥ n₄ + δ, adjoining this inequality with

dual variable y3 yields a dual solution {y2 = 0, y3 = 1₃} of cost ₁₂n + δ₃, increasing the bound

when δ ≥ 3. So dn₄e ≤ α7,5 < n₄ + 3. Because all graphs in the decomposition have six or seven

edges, α7,5 ≡ 0 (mod 3). Thus when n ≡ 9, 12 (mod 24), α7,5 ≡ 3 (mod 6), violating (4). This

increases the bound by 1 when n ≡ 9, 12 (mod 24).

Case 3: n ≡ 2 (mod 3): Again consider the inequality from (6), and let y2 be its dual variable.

Each graph G`,q,p leads to the dual inequality ψ`,q,py2 ≤ p − 2₃q. The dual optimum of 2n₂₁ arises

when y₂? = ₂₁2 ; the only graph whose dual inequality is binding is G1,7,5 with ψ1,7,5 = 7₂ and

5 − 2₃7 = 1₃. We can compute the slackness of each variable; for α`,q,p, the slackness is p − 2

3q − 2

21ψ`,q,p. The only variables with slackness at most 4

7 are α2,7,5 and α3,7,5 with slackness 1 7,

α4,7,5 with slackness 2₇, and α1,5,4 with slackness 4₇. An increase of 4₇ would result in an increase

in the integer ceiling when n ≡ 2, 11, 14, 20 (mod 21), so in these cases we are restricted to K4s and graphs in {G`,7,5} to meet the bound. To satisfy (6), α7,5 ≥ d2n₇ e. If α7,5 ≥ 2n₇ + δ,

adjoining this inequality with dual variable y3 yields a dual solution {y2 = 0, y3 = 1₃} of cost 2n

21 + δ

3, increasing the bound when δ ≥ 3. So d

2n

7 e ≤ α7,5 < 2n

7 + 3. Because all graphs in

the decomposition have six or seven edges, α7,5 ≡ 1 (mod 3). Thus when n = 21s + x for

x ∈ {2, 11, 14, 20}, α7,5 = 6s + 1, 6s + 4, 6s + 4, 6s + 7, respectively. This violates (4) precisely

when n ≡ 44, 65; 11, 74; 14, 35; 20, 41 (mod 84), increasing the bound by 1 in these cases. We denote by L(7, n) the lower bound prescribed by Theorem 2.1.

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3 Group Divisible Designs with Block Size Four

A group divisible design (GDD) is a triple (X, G, B) where X is a set of points, G is a partition of X into groups, and B is a collection of subsets of X called blocks such that any pair of distinct points from X occur together either in some group or in exactly one block, but not both. A K-GDD of type gu1₁ g₂u2. . . g_sus is a GDD in which every block has size from the set K and in which there are ui groups of size gi, i = 1, 2, . . . , s.

A group divisible design (X, G, B) is resolvable if its block set B admits a partition into parallel classes, each parallel class being a partition of the point set X.

A pairwise balanced design (PBD) with parameters (K; v) is a K-GDD of type 1v.

The interested reader may refer to [6, 9] for the undefined terms as well as a general overview of design theory. The main recursive construction that we use is Wilson’s Fundamental Construction (WFC) for GDDs (see, e.g. [9]).

Construction 3.1 Let (X, G, B) be a GDD, and let w : X → Z+_{∪ {0} be a weight function on}

X. Suppose that for each block B ∈ B, there exists a K-GDD of type {w(x) : x ∈ B}. Then there

is aK-GDD of type {P

x∈Gw(x) : G ∈ G}.

A double group divisible design (DGDD) is a quadruple (X, H, G, B) where X is a set of points, H and G are partitions of X (into holes and groups, respectively) and B is a collection of subsets of X (blocks) such that

(i) for each block B ∈ B and each hole H ∈ H, |B ∩ H| ≤ 1, and

(ii) any pair of distinct points from X which are not in the same hole occur either in some group or in exactly one block, but not both.

A K-DGDD of type (g1, hv1)u1(g2, h2v)u2. . . (gs, hvs)us is a double group-divisible design in which

every block has size from the set K and in which there are ui groups of size gi, each of which

intersects each of the v holes in hi points. (Thus, gi = hiv for i = 1, 2, . . . , s. Not every DGDD

can be expressed this way, of course, but this is the most general type that we require.) Thus, for example, a modified group divisible design K-MGDD of type gu _{is a K-DGDD of type (g, 1}g₎u_.

A k-DGDD of type (g, hv)kis an incomplete transversal design ITD (k, g; hv) and is equivalent to a set of k − 2 holey MOLS of type hv (see, e.g. [9]). A DGDD is resolvable if its block set admits a partition into parallel classes. We use the following existence result.

Theorem 3.2 [20] There exists a 4-DGDD of type (mt, mt)n if and only if t, n ≥ 4 and (t − 1)(n − 1)m ≡ 0 (mod 3) except for (m, n, t) = (1, 4, 6) and except possibly for m = 3 and (n, t) ∈ {(6, 14), (6, 15), (6, 18), (6, 23)}.

We also make use of the following simple construction for 4-GDDs:

Construction 3.3 [19] Suppose that there is a 4-DGDD of type (g1, hv1)u1 (g2, hv2)u2. . . (gs,

hv_s)usand that for eachi = 1, 2, . . . , s there is a 4-GDD of type hv_ia1wherea is a fixed non-negative integer. Then there is a4-GDD of type hv_a1 _where_{h =}

s

X

i=1

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The following results on TDs are known. Theorem 3.4 A TD(k, m) exists if:

1. k = 5 and m ≥ 4 and m 6∈ {6, 10};

2. k = 6 and m ≥ 5 and m 6∈ {6, 10, 14, 18, 22};

3. k = 7 and m ≥ 7 and m 6∈ {10, 14, 15, 18, 20, 22, 26, 30, 34, 38, 46, 60, 62}. Finally, we employ the following results on 4-GDDs.

Theorem 3.5 ([9, III.1.3 Theorem 1.28]) A 4-GDD of type 3u_m1_{exists if and only if either}_{u ≡ 0}

mod 4 and m ≡ 0 mod 3, 0 ≤ m ≤ (3u − 6)/2; or u ≡ 1 mod 4 and m ≡ 0 mod 6, 0 ≤ m ≤ (3u − 3)/2; or u ≡ 3 mod 4 and m ≡ 3 mod 6, 0 < m ≤ (3u − 3)/2.

Theorem 3.6 ([17, Theorem 1.7]) There exists a 4-GDD of type g4m1 withm > 0 if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g₂.

Theorem 3.7 ([18, Theorem 1.6]) There exists a 4-GDD of type 6um1 for everyu ≥ 4 and m ≡ 0 mod 3 with 0 ≤ m ≤ 3u − 3 except for (u, m) = (4, 0) and except possibly for (u, m) ∈ {(7, 15), (11, 21), (11, 24), (11, 27), (13, 27), (13, 33), (17, 39), (17, 42), (19, 45), (19, 48), (19, 51), (23, 60), (23, 63)}.

Theorem 3.8 ([16, Theorem 3.16]) There exists a 4-GDD of type 12u_m1 _{for each} _{u ≥ 4 and}

m ≡ 0 mod 3 with 0 ≤ m ≤ 6(u − 1).

Theorem 3.9 ([16, Theorem 5.21]) There exists a 4-GDD of type 2u_m1 _{for each} _{u ≥ 6, u ≡ 0}

mod 3 and m ≡ 2 mod 3 with 2 ≤ m ≤ u − 1 except for (u, m) = (6, 5) and possibly excepting (u, m) ∈ {(21, 17), (33, 23), (33, 29), (39, 35), (57, 44)}.

3.1 g ∈ {24, 84}

Lemma 3.10 For each u ≥ 4, u 6∈ {7, 11, 13, 17, 19, 23}, there exists a 4-GDD of type 24u_m1_with

m ≡ 0 mod 3 and 0 ≤ m ≤ 12(u − 1).

Proof: For u = 4, see Theorem 3.6. For each u ≥ 5, u 6∈ {7, 11, 13, 17, 19, 23}, take a 4-GDD of type 6uv1 with v ≡ 0 mod 3 and 0 ≤ v ≤ 3(u − 1), and remove the points on the last group

of size v; apply weight 4, using 4-MGDDs of type 44 and resolvable {3}-MGDDs of type 43,

to obtain a {3, 4}-DGDD of type (24, 64₎u _{whose triples fall into 3v parallel classes. Adjoin 3v}

infinite points to complete the parallel classes and then fill in 4-GDDs of type 6ut1 with t ≡ 0 mod 3 and 0 ≤ t ≤ 3(u − 1) to obtain a 4-GDD of type 24u_{(3v + t)}1_{, as desired.}

Lemma 3.11 For each u ∈ {7, 11, 13, 17, 19, 23}, there exists a 4-GDD of type 24um1withm ≡ 0 mod 3 and 3(u − 1) ≤ m ≤ 12(u − 1).

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Proof: For each u, start with a TD(5, u) and adjoin an infinite point ∞ to the groups, then delete a finite point so as to form a {5, u + 1}-GDD of type 4u_u1_{. Note that each block of size u + 1}

intersects the group of size u in the infinite point ∞ and each block of size 5 intersects the group of size u, but certainly not in ∞. Now, in the group of size u, we give ∞ weight 0 or 3(u − 1) and give the remaining points weight 3, 6 or 9. Give all other points in the {5, u + 1}-GDD weight 6. Replace the blocks in the {5, u + 1}-GDD by 4-GDDs of types 6u, 6u(3(u − 1))1, 6431, 6461, or 64₉1_{to obtain the 4-GDDs as desired.}

Lemma 3.12 For each u ∈ {7, 11, 13, 17, 19, 23}, there exists a 4-GDD of type 24u_m1_with_{m ≡ 0}

mod 3 and 0 ≤ m ≤ 3(u − 2).

Proof: Starting from a 4-DGDD of type (24, 64₎u _{coming from Theorem 3.2 and applying}

Con-struction 3.3 with 4-GDDs of type 6um1 to fill in holes, we obtain most of the designs except for (u, m) ∈ {(7, 15), (11, 21), (11, 24), (11, 27), (13, 27), (13, 33), (17, 39), (17, 42), (19, 45), (19, 48), (19, 51), (23, 60), (23, 63)}.

For the remaining choices for (u, m), take a 4-GDD of type 6u31 and remove the points of the

last group of size 3; apply weight 4, using 4-MGDDs of type 44 and resolvable {3}-MGDDs of

type 43_{, to obtain a {3, 4}-DGDD of type (24, 6}4₎u_{whose triples fall into 9 parallel classes. Adjoin}

m − 9 infinite points to complete the parallel classes and then fill in 4-GDDs of type 6u(m − 9)1.

Combining Lemmas 3.10–3.12, we have the following theorem.

Theorem 3.13 There exists a 4-GDD of type 24u_m1 _{for each} _{u ≥ 4 and m ≡ 0 mod 3 with}

0 ≤ m ≤ 12(u − 1).

Theorem 3.14 There exists a 4-GDD of type 84um1 for each u ≥ 4 and m ≡ 0 mod 3 with

0 ≤ m ≤ 42(u − 1).

Proof: The proof is similar to that of Lemma 3.10. For each u, take a 4-GDD of type 12uv1 with v ≡ 0 mod 3 and 0 ≤ v ≤ 6(u − 1), and remove the points on the last group of size v; apply weight 7, using 4-MGDDs of type 74 _{and resolvable {3}-MGDDs of type 7}3_{, to obtain}

a {3, 4}-DGDD of type (84, 127)u whose triples fall into 6v parallel classes. Adjoin 6v infinite points to complete the parallel classes and then fill in 4-GDDs of type 12ut1 _{with t ≡ 0 mod 3}

and 0 ≤ t ≤ 6(u − 1) to obtain a 4-GDD of type 84u_{(6v + t)}1_{, as desired.}

4 Constructions: n ≡ 1 (mod 3)

We settle small cases first.

Lemma 4.1 A(7, n) = L(7, n) for n ∈ {4, 7}.

Proof: The lower bound is met for n = 4 by a single K4. The lower bound is realized when

n = 7: Let V = {∞} ∪ {0, . . . , 5}, and form the three G1,7,5s {{i, i + 3}, {i, i + 1}, {i, i + 4}, {i +

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Lemma 4.2 A(7, 10) = L(7, 10) + 1 = 32.

Proof: The lower bound of 31 is not met. To see this, the only primal variables with slackness at most 1₃ are for {G`,7,5}. But 6x + 7y = 45 and 4x + 5y = 31 admits only the solution x = 4

and y = 3, i.e. four K4s and three graphs from {G`,7,5}. There is a unique way to place four

K4s in a K10, and its complement does not partition into three graphs from {G`,7,5}. To produce a

decomposition of cost 32, on the 10 points {0, . . . , 9} form K4s on {0, 1, 2, 3} and {0, 4, 5, 6}, and

the graphs G2,7,5 {{2, 4}, {2, 5}, {2, 7}, {2, 9}, {4, 7}, {5, 7}, {4, 9}} G3,7,5 {{3, 9}, {5, 9}, {6, 9}, {7, 9}, {3, 6}, {3, 7}, {6, 7}} G4,7,5 {{3, 4}, {3, 5}, {3, 8}, {4, 8}, {5, 8}, {1, 4}, {1, 5}} G4,7,5 {{0, 7}, {0, 8}, {0, 9}, {7, 8}, {8, 9}, {1, 7}, {1, 9}} G1,5,4 {{1, 8}, {1, 6}, {2, 8}, {2, 6}, {6, 8}} Lemma 4.3 L(7, 19) + 1 ≤ A(7, 19) ≤ L(7, 19) + 2 = 117.

Proof: The lower bound of 115 cannot be met. A maximum packing on 19 points has 25 K4s [7].

Consider the linear program using (5). Using slackness, the only way to achieve a dual objective value of 1 in such a way that at least 21 = 19₂ − 25 · 6 edges do not appear in K4s is to use three

graphs in {G`,7,5}. There are 249 nonisomorphic graphs that can be left by a maximum packing

of 25 K4s in K19 [2]. G3,7,5 cannot be used because it contains a K4, and the 25 K4s form a

maximum packing. Of the 249 graphs, 79 have degree sequence 314; 122 have degree sequence

61₃12_{and 48 have degree sequence 6}2₃10_{. In order to use a G}

1,7,5there must be at least five vertices

of degree 6 or larger; and for G2,7,5 there must be at least three. Hence both are ruled out and the

only possibiiity is three G4,7,5s. This case can be eliminated by a simple computer search. Thus

the drop cost cannot be 115. A solution with drop cost 117 follows:

4 24 's: {0,1,2,4},{0,3,5,6},{0,7,8,9},{0,10,11,12},{0,13,14,15} {0,16,17,18},{1,3,7,10},{1,5,8,11},{1,6,13,16},{1,9,14,17} {1,12,15,18},{2,3,8,15},{2,5,9,18},{2,6,10,17},{2,7, K 12,13} {2,11,14,16},{3,4,14,18},{3,9,12,16},{4,5,12,17},{4,6,9,15} {5,10,15,16},{6,7,11,18},{6,8,12,14},{8,10,13,18} 2,7,5 one G :{{3,11},{3,13},{3,17},{11,15},{11,17},{13,17},{15,17}} 4,7,5 two G :{{4,7},{4,8},{4,16},{7,16},{7,17},{8,16},{8,17}} and {{4,10},{4,11},{4,13},{9,10},{9,11},{9,13},{11,13}} 7,6,6 one G :{{5,7},{5,13},{5,14},{7,14},{7,15},{10,14}}

Theorem 4.4 When n ≡ 1 (mod 3) and n 6∈ {10, 19}, A(7, n) = L(7, n). Moreover, A(7, 10) = L(7, 10) + 1 and L(7, 19) + 1 ≤ A(7, 19) ≤ L(7, 19) + 2.

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Proof: When n ≡ 1, 4 (mod 12), there is a 4-GDD of type 1n with drop cost L(7, n). When n ≡ 7, 10 (mod 12) and n 6∈ {10, 19}, there is a 4-GDD of type 1n−7₇1 _{[7]; fill the hole with a}

solution from Lemma 4.1.

5 Constructions: n ≡ 0 (mod 3)

The lower bound is met for n = 3 by a single K3.

Lemma 5.1 A(7, 6) = L(7, 6) + 1 = 12.

Proof: The lower bound of 11 is not met. A decomposition of cost 12 can be produced as follows: G2,7,5 {{0, 1}, {0, 2}, {0, 4}, {0, 5}, {1, 4}, {1, 5}, {2, 4}}

G2,7,5 {{1, 2}, {1, 3}, {2, 3}, {2, 5}, {3, 4}, {3, 5}, {4, 5}}

G1,1,2 {{0, 3}}

Lemma 5.2 A(7, 9) = L(7, 9) + 1 = 27.

Proof: The lower bound of 26 is not met for n = 9 as follows. There can be at most three K4s

on nine points. If there are zero, at least six graphs are needed each having slackness at least 1₃; because the total increase in the dual objective function is 2, all graphs must be from {G`,7,5} and

cannot account for 36 edges. In the same manner, with one K4, 30 edges must be accounted for by

graphs in {G`,7,5}, each with slackness 1₃ and G1,5,4 with slackness 2₃; again this is not possible as

25 is not a multiple of 7. There remain cases with two or three K4s; each can be eliminated by an

exhaustive search.

A decomposition of cost 27 using graphs on at most six edges is given in [2]. We give a different solution here:

G1,7,5 {{0, 7}, {0, 8}, {1, 7}, {1, 8}, {2, 7}, {2, 8}, {7, 8}} G4,7,5 {{0, 4}, {0, 5}, {0, 6}, {1, 4}, {1, 5}, {1, 6}, {4, 5}} G4,7,5 {{2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 6}} G4,7,5 {{4, 7}, {4, 8}, {5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}} G1,6,4 {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 3}} G1,2,3 {{3, 7}, {3, 8}} Lemma 5.3 A(7, 15) = L(7, 15) = 72.

Proof: Start with a Kirkman triple system of order 9 on {0, . . . , 8}, in which the first parallel class is {B0, B1, B2}. Then adjoin points {x0, x1, x2, y0, y1, y2}. Form nine K4s by adding yi to

each block of the (i + 2)nd parallel class. For i ∈ {0, 1, 2} form a K4on {xi+2} ∪ Bi and a G1,7,5in

which the degree 4 vertices are xiand xi+1and the degree 2 vertices are the elements of Bi. Form

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Lemma 5.4 A(7, 18) ≤ L(7, 18) + 1 = 105.

Proof: Form a 4-GDD of type 35 with groups {Bj : j = 0, 1, 2, 3, 4}. Then adjoin points

{x0, x1, x2}. For i ∈ {0, 1, 2}, form a G1,7,5 by using the edge {xi, xi+1 mod 3} and joining these

vertices to each vertex in Bi and form a K4 by adding xi+2 mod 3 to Bi. For i ∈ {3, 4}, form a

G3,6,5 by joining the vertices x0 and x1 to vertices in Bi and form a K4 by adding x2 to Bi. This

decomposition is of cost 105.

Lemma 5.5 A(7, 24) = L(7, 24) = 186.

Proof: We give the solution on {0, 1, 2, 3, 4, 5, 6, 7} × Z3, writing element (i, j) as ij.

(00, 01, 10, 42) (00, 11, 50, 61) (00, 20, 31, 32) (00, 21, 51, 52)

(00, 22, 70, 72) (00, 60, 62, 71) (10, 11, 21, 70) (10, 22, 51, 61)

(10, 31, 50, 71) (10, 32, 41, 62) (20, 21, 42, 61) (30, 50, 62, 72)

(40, 41, 52, 72) (30, 40 : 00, 10, 20) (30, 41 : 51, 61, 71)

The latter two orbits are graphs isomorphic to G1,7,5.

Theorem 5.6 A(7, n) = L(7, n) when n ≡ 0 (mod 3), n 6≡ 18 (mod 24) and 1. n ≥ 96 when n ≡ 0, 3, 6, 9, 15 (mod 24);

2. n ≥ 276 when n ≡ 12 (mod 24);

3. n ≥ 309 when n ≡ 21 (mod 24).

L(7, n) ≤ A(7, n) ≤ L(7, n) + 1 when n ≡ 18 (mod 24) and n ≥ 114.

Proof: If m = n mod 24 ∈ {0, 3, 6, 9, 15, 18} and n ≥ 96, form a 4-GDD of type 24(n−m)/24_m1

from Theorem 3.13; place optimal groomings from Lemma 5.5 on each group of size 24, and an optimal grooming of size m on the exceptional group (from Lemmas 5.1, 5.2, or 5.3 when m = 6, 9, 15, respectively). When m = 18, use the grooming from Lemma 5.4, missing the lower bound by 1. When m = 6, reduce the drop cost by 1 by amalgamating the single edge from this grooming with a K4 of the 4-GDD to form a G3,7,5. When m = 9, reduce the drop cost by 1 by

amalgamating both edges of the G1,3,2 of this grooming with K4s of the 4-GDD to form G3,7,5s.

When m = n mod 24 = 12, form a 4-GDD of type 204, and add four infinite points. On each group, together with the four infinite points, place an optimal grooming from Lemma 5.5 aligning a K4 on the four infinite points. Suppress the duplicate K4s so produced. This establishes that

L(7, 84) = A(7, 84). Then filling groups in a 4-GDD of type 24t₈₄1 _{establishes that A(7, 24t +}

84) = L(7, 24t + 84) when t ≥ 8, i.e. for all n ≥ 276.

When m = n mod 24 = 21, form a 4-GDD of type 234_{, and add one infinite point. On}

each group, together with the infinite point, place an optimal grooming from Lemma 5.5. This establishes that L(7, 93) = A(7, 93). Then filling groups in a 4-GDD of type 24t931 _establishes

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6 Constructions: n ≡ 2 (mod 3)

Lemma 6.1 A(7, n) = L(7, n) for n ∈ {5, 8}.

Proof: For K5, note that G1,7,5≡ K5\ K3. Partition K8 as follows:

G1,7,5 {{0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}}

G1,7,5 {{6, 7}, {6, 2}, {6, 3}, {6, 4}, {7, 2}, {7, 3}, {7, 4}}

G3,7,5 {{1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}}

G4,7,5 {{1, 6}, {1, 7}, {0, 5}, {0, 6}, {0, 7}, {5, 6}, {5, 7}}

Lemma 6.2 A(7, 11) = L(7, 11) = 39.

Proof: Partition K11 on {∞1, ∞2} ∪ (Z3 × Z3) as follows. Include the K4 {∞2, 02, 12, 22}.

Form three G2,7,5s as {{i0, (i + 1)1}, {i0, (i + 2)1}, {i0, (i + 1)2}, {i0, (i + 2)2}, {(i + 1)1, (i +

2)1}, {(i + 1)1, (i + 2)2}, {(i + 2)1, (i + 1)2}} for i ∈ {0, 1, 2}. Then include three G3,7,5s as

{{∞1, i0}, {∞1, i1}, {∞1, i2}, {i0, i1}, {i0, i2}, {i1, i2}, {∞2, i1}} for i ∈ {0, 1, 2}. Include one

last G3,7,5: {{∞1, ∞2}, {∞2, 00}, {∞2, 10}, {∞2, 20}, {00, 10}, {00, 20}, {10, 20}}.

Lemma 6.3 A(7, 17) ≤ L(7, 17) + 1 = 94.

Proof: Start with an S(2, 4, 16) on Z15∪ {∞} with blocks {i, i + 1, i + 3, i + 7} for i ∈ Z15and

{∞, i, i + 5, i + 10} for i ∈ {0, 1, 2, 3, 4}. We adjoin a new point α and modify six of the blocks in the first orbit as follows:

Block Remove Add

{5, 6, 8, 12} {8, 12} {α, 5}, {α, 8} {7, 8, 10, 14} {8, 14} {α, 7}, {α, 10} {0, 8, 9, 11} {0, 8} {α, 0}, {α, 9} {3, 11, 12, 14} {12, 14} {α, 3}, {α, 12} {0, 4, 12, 13} {0, 12} {α, 4}, {α, 13} {0, 2, 6, 14} {0, 14} {α, 2}, {α, 14}

Now add the K4 on {0, 8, 12, 14}. Then delete the K4 on {∞, 1, 6, 11}; on {α, ∞, 1, 6, 11}, place

a K3 and a G1,7,5. The result has 14 K4s, one K3, and seven graphs in {G`,7,5}.

Lemma 6.4 When n ≡ 2 (mod 6) and n ≥ 14, A(7, n) ≤ 2₃ n₂ + n₆ = ₃2 n₂ + 2n₂₁ +₁₄n.

Proof: Write h = n₂. When h ≡ 1 (mod 3) and h ≥ 7, a 4-GDD of type 2h exists by Theorem

3.9. It has h groups and h(h−1)₃ blocks. For each group, choose a distinct block containing one point of the group (this is an easy exercise using systems of distinct representatives). Then adjoin the pair of each group to its corresponding block to obtain a G3,7,5.

Lemma 6.5 When n ≡ 5 (mod 6) and n ≥ 23, A(7, n) ≤ 2₃ n₂ + 2n₂₁ +n+7 14 .

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Proof: Write h = n−5₂ . When h ≡ 0 (mod 3) and h ≥ 9, a 4-GDD of type 2h51 _{exists by}

Theorem 3.9. For each group of size 2, choose a distinct block containing one point of the group and adjoin the pair of each group to its corresponding block to obtain a G3,7,5. Then fill the group

of size 5 using a solution from Lemma 6.1.

In order to treat larger cases, we now develop a recursion.

Lemma 6.6 There exists a decomposition of K21into nine partial parallel classes ofK3s, and six

G1,7,5s.

Proof: We present a solution on {0, 1, . . . , 20} with rows as partial parallel classes:

0 2 13 1 12 15 9 14 17 3 10 20 4 5 19 7 11 16 6 8 18 0 18 20 1 2 16 11 17 19 3 12 13 4 7 8 6 9 10 5 14 15 0 1 11 13 17 18 3 9 16 4 12 14 7 10 19 2 5 6 0 3 5 1 8 17 4 13 16 7 9 20 6 11 15 2 10 14 0 8 14 1 5 20 2 3 17 4 10 15 6 13 19 11 12 18 0 9 15 1 13 14 3 18 19 4 6 20 2 7 12 5 8 16 0 10 16 1 9 19 12 17 20 3 8 15 2 4 11 5 7 18 0 12 19 1 10 18 15 16 17 6 7 14 2 8 9 11 13 20 5 10 17 3 11 14 4 9 18 7 13 15 6 12 16 8 19 20

The remaining edges partition into six G1,7,5s: {{7i + j, 7i + j + 2}, {7i + j, 7i + 4}, {7i +

j, 7i + 5}, {7i + j, 7i + 6}, {7i + j + 2, 7i + 4}, {7i + j + 2, 7i + 5}, {7i + j + 2, 7i + 6}} for j ∈ {0, 1} and i ∈ {0, 1, 2}.

We denote by X(n) the excess over the lower bound, i.e. X(n) = A(7, n) − L(7, n).

Theorem 6.7 Let (V, G, B) be a resolvable group-divisible design of type 7n, in which the blocks ofB are partitioned into parallel classes P1, . . . , Ps, and for1 ≤ i ≤ s every block of Pi has size

ki. Suppose that, for1 ≤ i ≤ s, a 4-GDD of type 3kiσ1i exists, and that

Ps i=1σi > 0. Then A(7, 21n + 8 + s X i=1 σi) ≤ L(7, 21n + 8 + s X i=1 σi) + X(8 + s X i=1 σi).

Proof: Suppose without loss of generality that σ1 > 0. Give weight three to each point of the

GDD (V, G, B). For 2 ≤ i ≤ s, adjoin σi new infinite points, and place a 4-GDD of type 3kiσ1i on

the inflation of each block of Pitogether with these infinite points. Then proceed similarly for P1,

but adding only σ1− 1 infinite points; in the 4-GDD, delete one point in the group of size σ1 to

form a {3, 4}-GDD of type 3k1(σ1 − 1)1 in which the blocks of size three form a (frame) parallel

class on the 3ki points. On each inflation of a group form a copy of the 21-point design from

Lemma 6.6. The nine partial parallel classes of blocks of size 3 formed can be completed to nine parallel classes on the 21n points using the triples from the {3, 4}-GDDs. Finally add nine further infinite points and extend each of the nine parallel classes to K4s using these infinite points. The

resulting design has a hole on the 8 +Ps

i=1σi infinite points added in total, which can be filled

with a solution of cost A(7, 8 +Ps

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Corollary 6.8 1. X(92) ≤ X(29).

2. Forn ∈ {11, 14, 17, 20, 23, 26, 29}, X(84 + n) ≤ X(n). 3. Forn ∈ {14, 20, 26, 32, 38, 44, 50}, X(105 + n) ≤ X(n).

4. For29 ≤ n ≤ 71 and n ≡ 2 (mod 3), X(147 + n) ≤ X(n).

Proof: Apply Theorem 6.7 using an RTD(k, 7) with k = 3, 4, 5, 7 as a resolvable GDD of type 7k_{with s = 7 and k}

1 = · · · = k7 = k.

Corollary 6.9 1. For29 ≤ n ≤ 80 and n ≡ 2 (mod 3), X(168 + n) ≤ X(n).

Proof: Apply Theorem 6.7 using an RTD(7, n) with n = 8, 9, 11, 13, 16 as a resolvable GDD of type 7nwith s = n and k1 = · · · = kn−1 = 7 and kn = n.

Theorem 6.10 For x ≥ 4, 0 ≤ m ≤ 42(x−1), m ≡ 0 (mod 3), and r ∈ {11, 14, 17, 20, 23, 26, 29}, A(7, 84x + m + r) ≤ L(7, 84x + m + r) + X(m + r).

Equivalently,X(84x + m + r) ≤ X(m + r).

Proof: Form a 4-GDD of type 84xm1 from Theorem 3.14. Adjoin r infinite points, and place a solution on each group of size 84 together with the r points, leaving a hole on the r points (from Lemma 6.8(2)). On the m + r points, place a solution with excess X(m + r).

Theorem 6.11 For m ≡ 2 (mod 3) and 2 ≤ m ≤ 83, L(7, 84x + m) ≤ A(7, 84x + m) ≤ L(7, 84x + m) + X(84x + m), where X(84x + m) is given in Table 1 (using the final bold entry for X(84x + m) in the row for m when the table does not provide a value). In particular, A(7, 84x + m) ≤ L(7, 84x + m) + 4 when 84x + m > 1094.

Proof: Apply Lemmas 6.1, 6.2, and 6.3 for x = 0 and m ∈ {5, 8, 11, 17}; then apply Lemmas 6.4 and 6.5 to provide an upper bound on X(84x + m) in general. Now apply Corollary 6.8 and 6.9 to improve these upper bounds. Finally apply Theorem 6.10.

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x m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 2 1 6 12 18 2 6 6 6 6 6 6 6 6 2 5 0 6 12 4 4 6 6 6 6 6 4 8 0 2 2 18 24 2 11 0 0 12 4 0 14 0 0 2 18 0 17 1 1 13 5 1 20 0 0 2 2 0 23 2 2 14 6 2 26 1 1 3 3 1 29 2 32 2 8 2 4 2 35 2 0 2 20 0 38 2 8 2 4 2 41 2 0 2 20 0 44 2 8 2 4 2 47 3 1 3 21 1 50 3 9 3 5 3 53 4 2 2 22 4 2 56 4 10 4 6 4 59 4 2 2 22 4 2 62 4 10 4 6 4 65 4 2 2 2 4 2 68 4 10 4 6 4 71 5 3 3 3 5 3 74 4 10 4 0 4 4 4 4 4 4 4 4 0 77 6 12 4 4 6 6 6 6 6 4 80 5 11 5 1 5 5 5 5 5 5 5 5 1 83 6 12 4 4 6 6 6 6 6 4

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7 Conclusions

Grooming with ratio 7 corresponds to the smallest ratio C for which optimal groomings do not consist primarily of C-edge graphs. Consequently, optimal grooming focusses on packings with K4s in this case. Despite this, the structures of the edges not appearing in K4s appear to exhibit

patterns that repeat modulo 12, 24, and 84 when n ≡ 1, 0, 2 (mod 3), respectively. In the lat-ter case techniques for constructing optimal groomings in all cases would necessitate the direct construction of many ‘small’ groomings. Therefore in this paper, we have instead found near-optimal groomings in which the construction deviates from the lower bound by a fixed constant independent of n. When n ≡ 0, 1 (mod 3), much more complete characterizations are given. Our conjecture is that, with few small exceptions, the lower bound proved here provides the correct cost of an optimal grooming.

Acknowledgments

This work has been partially funded by NSC-94-2115-M009-017 (Fu), the National Natural Sci-ence Foundation of China under Grant No. 10771193 (Ge), Zhejiang Provincial Natural SciSci-ence Foundation of China under Grant No. R604001 (Ge), and Program for New Century Excellent Talents in University (Ge), and by NSC-96-2115-M239-002 (Lu).

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