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What else can we apply this wonderful outcome? b0C. C. Lindner s comment  Many smart combinatorists who devoted themselves to be  graph theorist , that is good. I also know a combinatorist who can be a very good graph theorist and he decided to apply graph theory in constructing combinatorial designs, he is the cleverest one! Salute  Alex Rosa . (I shall explain his idea later in this talk.) GH< L My experiencefSince I become a faculty member of National Chiao Tung Univ. in 1987, I have been working on graph theory, mainly graph decomposition, graph coloring and related topics until 1995 when I heard the comment by Curt about working on designs. Then, everything is Decomposition! After I know Group Testing, I have more confidence to say: Decomposition is great! Lg<33- / PreliminariesA graph G is an ordered pair (V,E) where V the vertex set is a nonempty set and E the edge set is a collection of subsets of V. In the collection E, a subet (an edge) is allowed to occur many times, such edges are called multi-edges. If both V and E of G are finite, the graph G is a finite graph. G is an infinite graph otherwise. If E contains subsets which are not 2-element subsets, then G is a hypergraph. If all edges in E are of the same size k, then the graph is a k-uniform hypergraph.Z> J Continued & A simple graph is a 2-uniform hypergraph without multi-edges. A multi-graph is a 2-uniform hypergraph. A complete simple graph on v vertices denoted by Kv is the graph (V,E) where E contains all the 2-element subsets of V. Hence, Kv has v(v-1)/2 edges. We shall use lKv to denote the complete multi-graph with multiplicity l , I.e. each edge occurs l times.hZ N ! 6b 2 3M"Y Graph DecompositionVWe say a graph G is decomposed into graphs in H if the edge set of G, E(G), can be partitioned into subsets such that each subset induces a graph in H. For simplicity, we say that G has an H-decomposition. If H = {H}, then we say that G has an H-decomposition denoted by H|G. An H-decomposition of Kv is also known as an H-design of order v.W.e%>33)33+* (Balanced Incomplete Block Designs (BIBD)A BIBD or a 2-(v,k,l) design is an ordered pair (X,B) where X is a v-set and B is a collection of k-element subsets (blocks) of X such each pair of elements of X occur together in exactly l blocks of B. A Steiner triple system of order v, STS(v), is a 2-(v,3,1) design and it is well-known that an STS(v) exists iff v is congruent to 1 or 3 modulo 6. bn 8&  Another point of viewzThe existence of an STS(v) is equivalent to the existence of a K3-decomposition of Kv, i.e. decomposing Kv into triangles.H{@,S   More General The existence of a 2-(v,k,l) design can be obtained by finding a Kk-decomposition of lKv. Example: 2K4 can be decomposed into 4 triangles (1,2,3), (1,2,4), (1,3,4) and (2,3,4). A 2-(4,3,2) design exists and its blocks are: {1,2,3}, {1,2,4}, {1,3,4} and {2,3,4}. ^( ,AN?Pairwise Balanced DesignsF *If lKv can be decomposed into complete subgraphs of order in a prescribed set K, then we have a 2-(v,K,l) design, also known as a (v,K,l) pairwise balanced design(PBD). A (22,{4,7},1) PBD exists. A pair of orthogonal latin squares of order 22 can be constructed from this PBD! ZZaf(!33$P! YG7Group Divisible DesignsF A graph G is a complete m-partite graph if V(G) can be partitioned into m partite sets such that E(G) contains all the edges uv where u and v are from different partite sets. If the partite sets of G are of size n1, n2, & , nm, then the graph is denoted by K(n1,n2,& ,nm). In case that all partite sets are of the same size n, then we have a balanced complete m-partite graphs denoted by Km(n). A Kk-decomposition of Km(n) is a k-GDD and a l-fold k-GDD can be defined accordingly. (See it?)Z   "   x   (,} ]M>k-GDD with Specified TypesIf the group size of a GDD is replaced with groups of different sizes t1, t2, & , tm, then we have a k-GDD with type t1 t2 & tm. The GDD defined on Km(n) is of type nm. A GDD of type nm is a (mn,{m,n},1) PBD. To determine the possible types of 3-GDD is far from being solved. (All groups of the same size is constructed by H. Hanani.)QZG"   ,GDD with two associatesA group divisible design with two associates l1 and l2, GDD(n,m;k;l1,l2), is a design (X,G,B) with m groups each of size n and (i) two distinct elements of X from the same group in G occur together in exactly l1 blocks of B and (ii) two distinct elements of X from different groups in G occur together in exactly l2 blocks of B. A k-GDD defined earlier as a Kk-decomposition of Km(n) is a GDD(n,m;k;0,1). A GDD(n,m;k;l1,l2) can be viewed as a Kk-decomposition of the union of m (l1Kn) s and a l2Km(n).zZ- ? "  ## ,fS9# Graph decomposition worksLet n, m, l2 1 and l1 0. Then a GDD(n,m;3;l1,l2) exists if and only if (1) 2 divides l1(n-1) + l2(m-1)n, (2) 3 divides l1mn(n-1) + l2m(m-1)n2, (3) if m = 2 then l1 l2n/2(n-1), and (4) if n = 2 then l2(m-1) l1. (By Fu, Rodger and Sarvate for n, m 3, and Fu and Rodger for all the remaining cases.) Results are in Ars Combin. and JCT(A) (1998) respectively. MZ6Z    *  $ ,Q $"t-(v,k,l) Designs Let lKv(t) denote the complete t-uniform hypergraph of order v with multiplicity l. Then lKv(t) has l edges. A t-(v,k,l) design is a Kk(t)-decomposition of lKv(t). A Steiner quadruple system of order v is a 3-(v,4,1) design. Note: Kv is Kv(2). ZZG ,I" '.K Cycle SystemsA cycle is a connected 2-regular graph. We use Ck to denote a cycle with k vertices and therefore Ck has k edges. If G can be decomposed into Ck s, then we say G has a k-cycle system and denote it by Ck | G. If Ck | Kv, then we say a k-cycle system of order v exists. A 3-cycle system of order v is in fact a Steiner triple system of order v. ZZ1 2 + 9    ~"}" Known Results8Ck | Kv if and only if Kv is k-sufficient. Let v be even and I is a 1-factor of Kv. Then Ck | Kv  I if and only if Kv  I is k-sufficient. After more than 40 years effort, the above two theorems have been proved following the combining results of B. Alspach et al. (2001, JCT(B))Z9 t7 )&4-Cycle SystemsA 4-cycle system of order v exists if and only if v 1 (mod 8). (Alex Rosa s idea.) Using a-labeling. A mapping j from V(G) into {0, 1, 2, & , |E(G)|} is an a-labeling if {|j(u) - j(v)| : uv is an edge of G} = {1, 2, 3, & , |E(G)|} and there exists a l such that for each uv in E(G), either j(u) l < j(v) or j(v) l < j(u). C4 has an a-labeling. (See it?) So are the cycles of length 4k. A labeling without the second condition is called a b-labeling or a graceful labeling. ZZ4 %3) 3 37fffffff3 33B PP4PI:A Beautiful Idea!nTheorem (Alex Rosa, 1966) If a graph G of size q has an b-labeling, then K2q+1 can be decomposed into copies of G. Proof. Use difference method! Theorem (A. Rosa) If a graph G of size q has an a-labeling, then K2pq+1 can be decomposed into copies of G. Proof. Now, we have p starters. 39(3C)H9 More 4-Cycle Systems   A 4-cycle system of the complete multipartite graph G exists if and only if G is 4-sufficient. In fact, finding the maximum packing of the complete multipartite graph is also possible. (Billington, Fu, and Rodger, JCD 9) It is also done for multigraphs. (G and C)., Z3 4 -  +(Pentagon SystemsCompare to 4-cycle systems or 3-cycle systems, the study of 5-cycle systems is harder. It takes a long while to find the necessary and sufficient conditions (?) to decompose a complete 3-partite graph into C5 s. (Billington et al.) Problem: Let H be a 2-regular subgraph of Kv such that v is and odd integer, v 5 and v(v-1)/2 - |E(H)| is a multiple of 5. Then Kv  H has a C5-decomposition. (Kv  H is 5-sufficient.) (*) It is done for C3, C4 and C6.&ZZ3/ $ #5 +b & V8,*Balanced Bipartite Designs|For experimental purpose, bipartite designs were introduced many years ago. Definition (BBD) A balanced bipartite design with parameter (u,v;k;l1,l2,l3) (defined on X Y), (X Y, B), is a Kk-decomposition of l1Ku l2Kv l3Ku,v where |X| = u and |Y| = v. Note: A pair of distinct elements from X (respectively Y) occurs together in l1 (respectively l2) blocks of B and two elements from different sets occur together in B exactly l3 blocks.RZL 3. 333i 7 0.A different approachReplace K3 with C4, then we have a bipartite 4-cycle design denoted by (u,v;C4;l1,l2,l3) BQD. (Q for quadrangle) It is quite complicate to find all BQD s, but it is possible to construct each of them. (It takes a long time to put them together.) AJC, 2005 Similar work on 4-cycle GDD with two associates was obtained earlier by Fu and Rodger. (Combin., Prob. and Computing, 2001) ~   5 }X1/ 4-cycle GDD \ Let n, m 1 and l1, l2 0 be integers. A 4-cycle (n,m;C4;l1,l2) GDD exists iff (1) 2 divides l1(n-1) + l2n(m-1), (2) 8 divides l1mn(n-1) + l2n2m(m-1), and if l2 = 0 then 8 divides l1n(n-1), (3) if n = 2 then l2 > 0 and l1 2(m-1)l2, and (4) if n = 3 then l2 > 0 and l1 3(m-1)l2/2 - d(m-1)/9, where d = 0 or 1 if l2 is even or odd respectively. NZ%Z  ),  "5     $X)*Q :0 Applications 3 Experimental Designs Group Testings DNA library Screening Scheduling Sharing Scheme Synchronous Optical Networks More & U>4d-Disjunct Matrices  vTheorem(Kautz and Singleton, IEEE Inform. 1964) A d-disjunct matrix can identify all positive clones if their number does not exceed d. Let (V, B) be a Steiner t-design with v elements and block size k. Let Mr be a binary matrix where the n columns are labeled by an arbitrary set of n blocks of (V, B), the rows by all r-subsets of V, and the cell (i, j) is 1 if and only if the label of row i is contained in the label of column j. Then &  3,R J3e>& A5An Application!Theorem (Fu and Hwang) For each r < t, Mr is a d-disjunct matrix with (*) n is the number of clones and is the number of tests. (**) In fact, packing with large n works well.nIZrZ3 (o $ ,) E6Library ScreeningIn DNA library screening, there are many oligonucleotides (clones) to be tested whether they are positive or negative. An oligonucleotide is a short string of nucleotides A, T, G and C. The goal of a DNA library screening is to identify all positive clones. Economy of time and costs require that the clones be assayed in groups. Each group is called a pool. If a pool gives a negative outcome, all clones are negative. On the other hand, if the pool is positive, at the second stage we test each clone individually. (Two-stage test!)>Ze3,)AF7Continued & In such screening, a microtiter plate, which is an arrar with size 812 or 1624, etc. is utilized and different clones are settled in each spot, called well, of the plate. wqw& The problem turns out to be decomposing Kn into Kr Kc  s. (Or good packings!)D  K3ff)  $,$t t(  G8 Main ResultsJK2 K3 case was settled by J. E. Carter (1989). K3 K3 case by Fu et al. J.S.P.I. (2003). K2 K4 case by Mutoh et al. SIAM J. Discrete Math. (2003). What s next?  (+ ( (% ( (     J;Scheduling via Edge-ColoringA proper k-edge-coloring of a graph G is an assignment the elements of {1, 2, 3, & , k} to the edges of G such that each edge receives a color and incident edges receive distinct colors. It is equivalent to a decomposition of G into k matchings. An equalized k-edge-coloring gives a  good scheduling of jobs! (We can always do it.)DM 3L ZK<Sharing Scheme via Latin Square>The existence of a latin square of order n is equivalent a decomposition of Kn,n,n into triangles. Here each partite set of Kn,n,n is labeled with 1, 2, 3, & , n. A critical set of a latin square plays the role of determining the square uniquely with as less entries as possible. Hopefully the number of entries is around n2/4. (Open) (Su Do Ku!) Split the entries of a critical set nicely creats a sharing scheme.ZM ,  fff f 33f3r3/8+;1Synchronous Optical Networks3Many current network infrastructures are based on the synchronous optical network(SONET). A SONET ring typically consists of a set of nodes connected an optical fiber in a undirectional ring topology. Ten minutes later & Consider grooming ratio C. We would like to find a decomposition of KN into subgraphs of size at most C with the total number of orders of subgraphs a  Minimum .pZR R3>p 5  <2C = 4N = 9 : A 4-cycle system of order 9 works. An H-design of order 9 where H is K4 - e - f also works. (1,2,3), (4,5,6), (7,8,9) (1,4,7), (2,5,8), (3,6,9) (1,5,9), (2,6,7), (3,4,8) (1,6,8), (2,4,9), (3,5,7) How about other N?O 33=3 The object:Decomposing the complete graph of order N into as many subgraphs H with max. ratio r = e(H) / o(H) as possible! (e(H) C.) For example, C = 6. Choose K4. (Almost done by Bermond et al. SIAM D.M.) C = 7, also K4. (Why?) C = 8. K5 - e - f. Note: Not necessarily be maximum packings.Z+Z23  ,PP>7 n]L= More & }It is your term to find them out, good luck to you and all of us. 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C. Lindners comment My experiencePreliminaries Continued KGraph Decomposition)Balanced Incomplete Block Designs (BIBD)Another point of view More GeneralPairwise Balanced DesignsGroup Divisible Designsk-GDD with Specified TypesGDD with two associatesGraph decomposition workst-(v,k,) DesignsCycle SystemsKnown Results4-Cycle SystemsA Beautiful Idea! More 4-Cycle Systems Pentagon SystemsBalanced Bipartite DesignsA different approach 4-cycle GDD Applicationsd-Disjunct MatricesAn Application!Library Screening Continued K Main ResultsScheduling via Edge-Coloring Sharing Scheme via Latin SquareSynchronous Optical NetworksC = 4 The objectMore K  ϥΦr ²]pdO OLE A{ vD%_hlfuhlfu  !"#$%&'(*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./012345679:;<=>?ABCDEFGIJKLMNOTRoot EntrydO)PicturesPCurrent UserHSummaryInformation(8PowerPoint Document()DocumentSummaryInformation8@