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)™Ê¦b~±s*0™Ú²IBž¦T<‡Ç¨WCvª¨Žvmtƒ‘�vy”œ›Žy]JSw—™�€yzŠŸ„{‰ˆyb{¬Î¦xTHQ\\j†­²•bHY‰£˜‚†š›vv†Š€wv…�†vpv€ŠŽ›˜‡vm}�ŠtaLq“Ÿ¤Šouorit“Ÿ�to}‰�z{||Œ��kks€Ž‹ˆˆzxwt�‹‡qi„ˆ…v`V`…”¢¦„Y]„š’ƒy‰Œ{p~ˆ~ƒ��€trrvyt~Šƒpoy‡eVgp†�˜€xnv”¦º¢“¯å�}žjVW;53"Sƒ©ÐâÖ¦k20Ml‰ª¿É±�—�~on’ Œoadgkqƒœ™Šlxcg—�’�}jm~gwnƒ“½ßħP9¿è¶bdYh�¡¨Îªsz^`XC?ÿ Yó2ð$¬½0/è6ög¯ßΛÁÿ&ùó2ð$õŽ¡}|n­ÓÛ»1êÿÛ ó2ð$�ñè.Ä•rƒtVŒµkÿŠ)ú#ó2ð$ð}›lÀž;[nXÁç/76ÿå„Mó2ð$p@õl8-ÜÉõ4ßý?§£ÿ_iOó³ ð‚�0e‚˜²ƒ0e„˜²…‡ˆ‰¿ ô   �ƒ¿ÀÁÂÿÿÿà ÄÅAÆÁÇÈÉÊËjJÌÍÎÏÁÐÑÒÓÔÕ×ÿËËË 8c8c     ?€�‚ƒœ1„…ðù†‡÷ˆ ¿ÀÁÂdÃÄÅÆÇÈÉÊ0uËÐÌ0íìÿÍ@T‰Î€Ï€ÿÿÐyÿÑ2Ò NÓPÃÔÕ'Öp”×°<ÿÿØÙ'Úp”ÿA¨)BCD|¾E„|¾…†|¾‡S"ñ��‘’¿‚‚`ñÿ3ff™Ìÿfÿ3ÌÿPPf@ñ÷ðó€Ð7ºuìÊš;2NÍÉÊš;úgþý4BdBdì 0ØØ ÿÿÿ ÿÿÿpûppû@ <ý4!d!d® 0”Ü�3Ú<ý4dddd® 0”Ü�3Úÿ ?Ù Ú%ðj†óŸ¨(Graph Decomposition and its ApplicationsŸ <Hung-Lin Fu (…PF`—) WËz¤N�'Yx[Éa(uxex[û|ªóŸ¨ MotivationŸ¨þThe study of graph decomposition has been one of the most important topics in graph theory and also play an important role in the study of the combinatorics of experimental designs (combinatorial designs). What else can we apply this wonderful outcome? ª�bóŸ 0C. C. Lindner s comment Ÿ Ž 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.) ¡GHª< LÏóŸ¨ My experienceŸ¨fSince 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<ÿ3þÿ3þª- /óŸ¨ PreliminariesŸ¨ñA 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.¡òZòª>™ó J óŸ Continued & Ÿ ÎA 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 lðKv to denote the complete multi-graph with multiplicity lð , I.e. each edge occurs lð times.¡¤hZ™ çÿN çÿ!‚ çÿ6‚‚ªb 2 3M"Yó Ÿ¨Graph DecompositionŸ¨VWe 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�’%�’�’>��ÿ3þ�ÿ3þçÿ)�ÿ3þÿ3þª+*ó Ÿ¨(Balanced Incomplete Block Designs (BIBD)Ÿ ÂA BIBD or a 2-(v,k,lð) 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 lð 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. ¡Žb‚��n‚ �—�ª8&ó  Ÿ¨Another point of viewŸ¨zThe existence of an STS(v) is equivalent to the existence of a K3-decomposition of Kv, i.e. decomposing Kv into triangles.¡H{@çÿçÿçÿª,Só  Ÿ¨ More GeneralŸ  The existence of a 2-(v,k,lð) design can be obtained by finding a Kk-decomposition of lðKv. 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}. ¡^(€ˆçÿ€ˆçÿ €ˆçÿ¡€ª,A¯óN?Ÿ¨Pairwise Balanced Designs¡Fª Ÿ *If lðKv can be decomposed into complete subgraphs of order in a prescribed set K, then we have a 2-(v,K,lð) design, also known as a (v,K,lð) 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! ¡ÊZZ€çÿa€€‚‚fþ(!ÿ3þÿ3þ$ªP! YG7óŸ¨Group Divisible Designs¡F Ÿ ÖA 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 lð-fold k-GDD can be defined accordingly. (See it?)¡ìZÖ çÿ çÿ çÿ" çÿ çÿ çÿx çÿ çÿ çÿ(‚�ª,} ]óM>Ÿ¨k-GDD with Specified TypesŸ  If 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.)¡ÔQZGçÿçÿçÿ"çÿ çÿ çÿçÿ •ª,Á„óŸ¨GDD with two associatesŸ êA group divisible design with two associates lð1 and lð2, GDD(n,m;k;lð1,lð2), 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 lð1 blocks of B and (ii) two distinct elements of X from different groups in G occur together in exactly lð2 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;lð1,lð2) can be viewed as a Kk-decomposition of the union of m (lð1Kn) s and a lð2Km(n).¡zöZ-‚Šçÿ‚Šçÿ ‚Šçÿ‚Šçÿ‚‚�‚ƒ‚�?‚‚‚‚‚‚‚Šçÿ ‚‚‚"‚‚‚‚‚‚‚Šçÿ ‚‚‚ ‚Šçÿ‚Šçÿ#‚Šçÿ‚Šçÿ‚Šçÿ#‚Šçÿ‚Šçÿ ‚Šçÿ‚Šçÿ‚ª,fS9ó# Ÿ¨Graph decomposition worksŸ Let n, m, lð2 ³ð 1 and lð1 ³ð 0. Then a GDD(n,m;3;lð1,lð2) exists if and only if (1) 2 divides lð1(n-1) + lð2(m-1)n, (2) 3 divides lð1mn(n-1) + lð2m(m-1)n2, (3) if m = 2 then lð1 ³ð lð2n/2(n-1), and (4) if n = 2 then lð2(m-1) ³ð lð1. (By Fu, Rodger and Sarvate for n, m ³ð 3, and Fu and Rodger for all the remaining cases.) Results are in Ars Combin. and JCT(A) (1998) respectively. ¡–MZ6Z ‚ çÿ ‚Šçÿ‚ ‚Šçÿ‚Šçÿ*‚Šçÿ ‚Šçÿ‚Šçÿ ‚Šçÿ‚Š‚Šçÿ‚Šçÿ$‚Šçÿ ‚Šçÿ�‚ª,ýQ $óŸ "t-(v,k,lð) Designs¡ €Ÿ üLet lðKv(t) denote the complete t-uniform hypergraph of order v with multiplicity lð. Then lðKv(t) has lð edges. A t-(v,k,lð) design is a Kk(t)-decomposition of lðKv(t). A Steiner quadruple system of order v is a 3-(v,4,1) design. Note: Kv is Kv(2). ¡øçZZ€çÿG €ˆçÿˆ,€ˆçÿˆ€ˆçÿˆI€ˆçÿ€ˆçÿˆ€ª†" '.KóŸ¨ Cycle SystemsŸ ²A 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. ¡šZZ1 çÿ2 çÿ+ çÿ9 çÿ  çÿ çÿ~ª"Ú}ó"Ÿ¨ Known ResultsŸ 8Ck | 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))¡’Zçÿçÿçÿ9çÿ çÿçÿçÿ¥ªt7 †ó)&Ÿ¨4-Cycle SystemsŸ ¾A 4-cycle system of order v exists if and only if v ºð 1 (mod 8). (Alex Rosa s idea.) Using að-labeling. A mapping jð from V(G) into {0, 1, 2, & , |E(G)|} is an að-labeling if {|jð(u) - jð(v)| : uv is an edge of G} = {1, 2, 3, & , |E(G)|} and there exists a lð such that for each uv in E(G), either jð(u) £ð lð < jð(v) or jð(v) £ð lð < jð(u). C4 has an að-labeling. (See it?) So are the cycles of length 4k. A labeling without the second condition is called a bð-labeling or a graceful labeling. ¡èÞZZ4 ‚%†ÿ3þ)†¦ †ÿ3þ‚¢ †3¦7¦fþ†fþ¦fþ†fþ††fþ¦fþ†fþ†‚Šçÿ‚†ÿ3þ ††ÿ3þ‚†ÿ3þB‚ †ÿPPþ‚ª4¼PÍóI:Ÿ¨A Beautiful Idea!Ÿ nTheorem (Alex Rosa, 1966) If a graph G of size q has an bð-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 að-labeling, then K2pq+1 can be decomposed into copies of G. Proof. Now, we have p starters. ¡ô„ˆÿ3þ9‚Šçÿ(‚†‚†ÿ3þC‚Šçÿ)‚†‚†‚óH9Ÿ¨ More 4-Cycle Systems Ÿ¨  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).¡, Z„ÿ3þ €ª4º -  ó+(Ÿ¨Pentagon SystemsŸ ŠCompare 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 ³ð 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.¡&èZÞZ�ÿ3þ/ çÿ�$ çÿ#5‚Šçÿ ‚Šçÿ‚Šçÿ+‚Šçÿ‚Šçÿ‚Šçÿ‚ªbÔ & V8ó,*Ÿ¨Balanced Bipartite Designs¡�Ÿ |For experimental purpose, bipartite designs were introduced many years ago. Definition (BBD) A balanced bipartite design with parameter (u,v;k;lð1,lð2,lð3) (defined on X Èð Y), (X Èð Y, B), is a Kk-decomposition of lð1Ku Èð lð2Kv Èð lð3Ku,v where |X| = u and |Y| = v. Note: A pair of distinct elements from X (respectively Y) occurs together in lð1 (respectively lð2) blocks of B and two elements from different sets occur together in B exactly lð3 blocks.¡R¿ZL ‚ÿ3þ�.‚Šçÿ‚Šçÿ‚Šçÿ‚ ‚‚‚ƒ‚†ÿ3þŽÿ3þçÿ†ÿ3þ‚Šçÿ‚Šçÿ‚Šçÿ‚Šçÿ‚Šçÿ‚Šçÿi‚Šçÿ‚Šçÿ ‚‚ƒ7‚‚ƒ ‚Šçÿ‚Šƒçÿª¿þó0.Ÿ¨A different approachŸ úReplace K3 with C4, then we have a bipartite 4-cycle design denoted by (u,v;C4;lð1,lð2,lð3) 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) ¡Ð~  çÿ çÿ5‚Šçÿ‚Šçÿ‚Šçÿ‚Šçÿ ‚ †}‚ªXó1/Ÿ¨ 4-cycle GDDª Ÿ \ Let n, m ³ð 1 and lð1, lð2 ³ð 0 be integers. A 4-cycle (n,m;C4;lð1,lð2) GDD exists iff (1) 2 divides lð1(n-1) + lð2n(m-1), (2) 8 divides lð1mn(n-1) + lð2n2m(m-1), and if lð2 = 0 then 8 divides lð1n(n-1), (3) if n = 2 then lð2 > 0 and lð1 £ð 2(m-1)lð2, and (4) if n = 3 then lð2 > 0 and lð1 £ð 3(m-1)lð2/2 - dð(m-1)/9, where dð = 0 or 1 if lð2 is even or odd respectively. ¡NŠZ%Z  ‚Šçÿ‚Šçÿ)‚Šçÿ‚Šçÿ‚Šçÿ,‚Šçÿ ‚Šçÿ‚Šçÿ‚Šçÿ ‚Šçÿ‚Š"‚Šçÿ‚Šçÿ5‚Šçÿ ‚Šçÿ ‚Šçÿ ‚Šçÿ ‚Šçÿ ‚Šçÿ$‚Šçÿ‚‚ªX)*Q æó:0Ÿ¨ Applications¡ ÿ3þª Ÿ îExperimental Designs Group Testings DNA library Screening Scheduling Sharing Scheme Synchronous Optical Networks More & ªUó>4Ÿ¨d-Disjunct Matricesª  Ÿ 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 & ¡ ¼‚ÿ3þ,‚R‚‚‚‚ çÿJÿ3þ‚‚eƒª>& “èóA5Ÿ¨An Application!Ÿ¨ºTheorem (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.¡nIZrZÿ3þ (çÿo $ ª,) €óE6Ÿ¨Library ScreeningŸ¨In 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!)¡>Ze‚ÿ3þ¢ª,)A“óF7Ÿ Continued & Ÿ In such screening, a microtiter plate, which is an arrar with size 8×12 or 16×24, etc. is utilized and different clones are settled in each spot, called well, of the plate. w•qŠíwªŠ& The problem turns out to be decomposing Kn into Kr × Kc  s. (Or good packings!)¡òD  K‚ÿ3þ‚�fþfþ)‚ ‚çÿ‚ ‚çÿ‚$‚,‚çÿ$‚‚ªt t(  óG8Ÿ¨ Main ResultsŸ JK2 × 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? ¡À¦çÿ (çÿ+ (çÿ (çÿ% (çÿ (çÿ  ‚  ‚ óJ;Ÿ¨Scheduling via Edge-ColoringŸ ˜A 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.)¡DMà ÿ3þLªê ZóK<Ÿ¨Sharing Scheme via Latin SquareŸ >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.¡¾ ZM çÿ, çÿ– fþfþfþ fþ 3ÿ3þfþÿ3þªr3/8ž+ó;1Ÿ¨Synchronous Optical Networks¡ÿ3ÌþŸ Many 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 .¡p�ZR‚‚ çÿRÿ3þª>­p 5  ó<2Ÿ¨C = 4Ÿ¨èN = 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?¡€éO çÿÿ3þÿ3þó=3Ÿ¨ The objectŸ :Decomposing the complete graph of order N into as many subgraphs H with max. ratio rð = eð(H) / oð(H) as possible! (eð(H) £ð 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.¡ÌóZ+Z2ÿ3þ€€ çÿ‚ ‚çÿçÿ ,ÿPPþª>7 n]óL=Ÿ  More & ¡Ÿ¨}It is your term to find them out, good luck to you and all of us. 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