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")$%&'(# '*+-,./01234 _šÜ2š$ńč.ĕrƒtVŒµk’Š)2š$aŗNÖhŠ`“Bųćįœś’“Š)2š$„ņm:lĮ8źŹßMTŠ>?’ >=2š$¬½0/č6ögÆßĪ›Į’&Ž?2š$sĄ€|"$ĢŲ¦ćĖ-`’LA³ š‚0e‚˜²ƒ0e„˜²…‡ˆ‰æ ō   ƒæĄĮĀ’’’Ć ÄÅAĘĮĒČÉŹĖjJĢĶĪĻĮŠŃŅÓŌÕ×’ĖĖĖ 8c8c     ?€‚ƒœ1„…šł†‡÷ˆ æĄĮĀdĆÄÅĘĒČÉŹ0uĖŠĢ0ķģ’Ķ@T‰Ī€Ļ€’’Šy’Ń2Ņ NÓPĆŌÕ'Öp”×°<’’ŲŁ'Śp”’AØ)BCD|¾E„|¾…†|¾‡S"ń‘’æ‚‚0ń ’3ff™Ģ’f@ń÷šó€Š„ŗuģŹš;2NĶÉŹš;śgžż4CdCdģ 0pڬ’’’¼ž’’pūppū@ <ż4!d!d® 0,Žx4<ż4dddd® 0,Žx4’ ˆfŠ^ŗ___PPT9‹@®8Æ ¬ €’’?Ł Ś%š!²óŸØ.Graph Decomposition vs. Combinatorial Design Ÿ <Hung-Lin Fu (…PF`—) WĖz¤N'Yx[Éa(uxex[ū|ŖóŸØ MotivationŸØ6The 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). Graph theorist can obtain more applications in combinatorial designs than graph decomposition its own. ŖšóŸ 6My advisor s comment (1995)Ÿ zFrom Curt Lindner (C.C. Lindner) : I have known 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 spent my sabbatical year 1994-1995 in Auburn University and I was lucky to hear the comment in a combinatorial seminar.”¾Z¾Ŗ,;LóŸØ My experienceŸ HSince 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. b_N ^g/fN Pp€fŗN. ¼e/f, bĶ‘°eQŽV†OxvzD}T-ŠŠ. FO/f, ŽVbfŠW„(uW„v‚iõ_†OTS©RU†t. ”%%Ŗ,- ø2óŸØ PreliminariesŸØńA 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.”ņZņŖ>™ó J óŸ Continued & Ÿ ĪA 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 lšKv to denote the complete multi-graph with multiplicity lš , I.e. each edge occurs lš times.”¤hZ™ ē’N ē’!‚ ē’6‚‚Ŗb 2 3M"Yó ŸØGraph DecompositionŸØVWe 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’%’’V ē’)Ŗ+*ó ŸØ(Balanced Incomplete Block Designs (BIBD)Ÿ ĀA BIBD or a 2-(v,k,lš) 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 lš 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‚ —Ŗ8&ó  ŸØAnother point of viewŸØzThe existence of an STS(v) is equivalent to the existence of a K3-decomposition of Kv, i.e. decomposing Kv into triangles.”H{@ē’ē’ē’Ŗ,Só  ŸØ More GeneralŸ  The existence of a 2-(v,k,lš) design can be obtained by finding a Kk-decomposition of lšKv. 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}. ”^(€ˆē’€ˆē’ €ˆē’”€Ŗ,AÆóŸØGroup Divisible DesignsŸ 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 lš-fold k-GDD can be defined accordingly. (See it?)”ģZÖ ē’ ē’ ē’" ē’ ē’ ē’x ē’ ē’ ē’(‚Ŗ,} ]óŸØGDD with two associatesŸ źA group divisible design with two associates lš1 and lš2, GDD(n,m;k;lš1,lš2), 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 lš1 blocks of B and (ii) two distinct elements of X from different groups in G occur together in exactly lš2 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;lš1,lš2) can be viewed as a Kk-decomposition of the union of m (lš1Kn) s and a lš2Km(n).”zöZ-‚Šē’‚Šē’ ‚Šē’‚Šē’‚‚‚ƒ‚?‚‚‚‚‚‚‚Šē’ ‚‚‚"‚‚‚‚‚‚‚Šē’ ‚‚‚ ‚Šē’‚Šē’#‚Šē’‚Šē’‚Šē’#‚Šē’‚Šē’ ‚Šē’‚Šē’‚Ŗ,fS9ó# ŸØGraph decomposition worksŸ Let n, m, lš2 ³š 1 and lš1 ³š 0. Then a GDD(n,m;3;lš1,lš2) exists if and only if (1) 2 divides lš1(n-1) + lš2(m-1)n, (2) 3 divides lš1mn(n-1) + lš2m(m-1)n2, (3) if m = 2 then lš1 ³š lš2n/2(n-1), and (4) if n = 2 then lš2(m-1) ³š lš1. (By Fu, Rodger and Sarvate for n, m ³š 3, and Fu and Rodger for all the remaining cases.) Results are in Ars Combin. and JCT(A) (1998) respectively. ”–MZ6Z ‚ ē’ ‚Šē’‚ ‚Šē’‚Šē’*‚Šē’ ‚Šē’‚Šē’ ‚Šē’‚Š‚Šē’‚Šē’$‚Šē’ ‚Šē’‚Ŗ,żP $óŸ "t-(v,k,lš) Designs” €Ÿ üLet lšKv(t) denote the complete t-uniform hypergraph of order v with multiplicity lš. Then lšKv(t) has lš edges. A t-(v,k,lš) design is a Kk(t)-decomposition of lšKv(t). A Steiner quadruple system of order v is a 3-(v,4,1) design. Note: Kv is Kv(2). ”ųēZZ€ē’G €ˆē’ˆ,€ˆē’ˆ€ˆē’ˆI€ˆē’€ˆē’ˆ€Ŗ†" '.Kó ŸØ EmbeddingsŸ TAn STS(u) can be embedded in STS(v) iff Kv  Ku has a K3-decomposition. A partial Steiner triple system of order u can be viewed as a subgraph H of Ku. Then H can be embedded in a Steiner triple system of order v iff Kv  H can be decomposed into triangles. It is conjectured that Kv  H can be decomposed into triangles if v > 2u and v ŗš 1 or 3 (mod 6). Note: H is an even graph with 3t edges for some non-negative integer. ”¼«Z) ē’ ē’ ē’] ē’D ē’? ē’6‚CŖP$[ G=Žó Ÿ 2U\Ŗ Ÿ The conjecture has been verified for several special classes of graphs H. If H is a complete graph, then it is the well-known Doyen and Wilson theorem. If H is corresponding to the maximum packing of Ku, then it is proved by Fu, Lindner and Rodger. The version of embedding lšKu  H in lšKv does have similar results. The case when H is corresponding to the maximum packing of lšKu was completely settled by Su, Fu and Shen recently after an earlier effort by Milici, Quattrocchi and Shen on the case when lš is even.”¶JZ¹’" ZJ€ €ē’H€‚€Š€ē’ ‚€Š€ē’Y‚€Š€璈‚€Ŗb€$ óŸ Continued & Ÿ The embedding problem of partial Steiner triple system has been considered for more than 30 years starting with a result by C.C. Lindner who proved that a partial Steiner triple system can be finitely embedded. The best result so far was proved by Hilton et al. that Kv  H can be decomposed into triangles for admissible v > 4u. They use edge-coloring technique to prove the result. Note: Darryn Bryant Mentioned recently that he can improve to v > 3u, but I am not able to locate the reference at this moment.”,Z  ē’õŖ, ytóŸØProblemŸ Kv  H ”. 2 ē’ŖóóŸØNecessary conditionsŸ <If Kv  H has a K3-decomposition, then the graph must have 3t edges for some t and each vertex is of even degree (even graph). Definition (x-sufficient): A graph G is said to be x-sufficient if x | |E(G)| and G is an even graph. If G has a K3-decomposition, then G is 3-sufficient.”n ZZē’ ē’m ƒ’3žhē’-ŖóŸØNash-Williams Conjecture(1970)Ÿ ˆ Let G be a 3-sufficient graph of order n and the minimum degree of G is not less than 3n/4. Then G has a K3-decomposition for sufficiently large n. Why 3n/4? (Dš(H) < n/4 where G = Kn  H.)”lÅZqē’* ’3ž„’3žŒ’3žē’„’3ž’3žŖ"¶óŸØ6Example: A graph G of order 24m+12 and valency 18m+8. Ŗ'ŸØO6m+3”&ē’óŸØ Known ResultsŸ öTheorem(C. Colbourn and A. Rosa, 1986) Let H be a 2-regular subgraph of Kv such that v is an odd integer not equal to 9 and v(v-1)/2 - |E(H)| is a multiple of 3. Then Kv  H has a K3-decomposition. Note: We can also consider the above theorem as packing Kv with K3 s such that the leave is H. Let H = C4 Čš C5. Then K9  H can not be decomposed into K3 s. (See it?)”^'ZUZF ē’^ ē’  璁5 ē’ ē’‚ ē’‚Šē’ ‚Šē’!‚Šē’‚‚‚Ŗb - ]XróŸ Continued & Ÿ XTheorem(Gustavsson, Ph.D. thesis 1991) Nash-Williams conjecture holds for the graphs which are 3-sufficient and minimum degree not less than (1  10-24)n. Note : I am not able to locate the reference of this result at this moment, the proof is very difficult to check. P.S.  POUL˜ÉarŠ g2U\„vzz“•.”Z'ZZ’oŖ$  óŸØ,Revised Version of Nash-Williams ConjectureŸ ōK3-packing Conjecture(2004) Let G be an even graph of order n and the minimum degree of G is not less 3n/4. Then, for sufficiently large n, G has a K3-packing with leave L where L is an empty graph, 4-cycle, or 5-cycle depending on the cases |E(G)| is congruent to 0, 1, or 2 modulo 3 correspondingly. First Test : Can we revise Colbourn and Rosa s result on quadratic leaves? ”˜ZßZ ē’ y ē’— CwŖR  ó!ŸØAn Idea works!ŸØAdjust the leave a little bit.óŸØProblemsŸ tLet v be an even integer and H be an odd spanning forest of Kv such that Kv  H is 3-sufficient. Then Kv  H has a K3-decomposition. (bgó`ć‰zl„vOUL˜.) Let v be an even integer and H be an odd spanning subgraph of Kv such that Dš(H) is at most 3 and Kv  H is 3-sufficient. Then Kv  H has a K3-decomposition.”Ę;=ē’ ē’&ē’ ē’ Aē’ €ē’ē’ ē’Ŗ˜< % 2 óŸ Continued & Ÿ ,Can we embed the K3-packings of Ku obtained by Colbourn and Rosa in a Steiner triple system of larger order v? Clearly, this result extend the work of embedding maximum packings of Ku with K3 s in triple systems when u is odd. We have more partial triple systems to embed now. ”dē’ē’•ē’ē’Vē’Ŗ>  r dóŸØ 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 ē’2 ē’+ ē’9 ē’  ē’ ē’~Ŗ"Ś}ó"ŸØ 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))”’Zē’ē’ē’9ē’ ē’ē’ē’„Ŗt7 †ó)&ŸØC3 C4”6ē’ē’Ÿ <A 4-cycle system of order v exists if and only if v ŗš 1 (mod 8). 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)”$4ꀪū ó%"ŸØPacking with 4-cyclesŸ *The maximum packing of Kv with C4 s has leave Li, i Īš Z8 for v ŗš i (mod 8) and Li is F, Ęš, F, C3, F, E6, F, C5 depeding on i = 0, 1, 2, & , 7. Similar result as Colbourn and Rosa s theorem: Let H be a 2-regular subgraph of Kv where v is odd. Then Kv  H has a C4-decomposition if and only if v(v-1)/2 - |E(H)| is a multiple of 4 (Kv  H is 4-sufficient). (Fu and Rodger, GC 2001) Surprisingly: If H is a spanning forest of Kv where v is even, then Kv  H has a C4-decomposition iff Kv  H is 4-sufficient. (Fu and Rodger, JGT 2000)”śZ ē’ ē’ ē’‚‚Šē’‚Šē’ ‚Šē’‚Šē’‚Šē’q‚Šē’‚Šē’ ‚Šē’E‚Šē’?‚‚Ёē’‚Ёē’ ‚Ёē’‚Ёē’‚‚Ŗ¼V ( ) O[0ó&#Ÿ Continued & Ÿ ōLet H be an odd graph with Dš(H) not greater than 3. Then Kv  H has a C4-decomposition if and only if Kv  H is 4-sufficient except two special cases when v = 8. (C.M. Fu, Fu, Rodger and Smith, DM 2004) Conjecture(Fu) Let H be a subgraph of Kv with Dš(H) £š v/4 and 4 £š k £š v. Then Kv  H has a Ck-decomposition if and only if Kv  H is k-sufficient. š Why v/4?”ŽZZ ‚Šē’ ‚Šē’‚Šē’d‚ ‚*‚Šē’'‚Šē’ ‚Šē’‚Šē’‚ †’3žŖt:*Š %*#ó'$ŸØ An example for k = 4Ÿ XK8  H can not be decomposed into 4-cycles. ”,-€ˆē’+€Ŗ+ó*'ŸØAnother EvidenceŸ hLet H be a 2-regular subgraph of Kv. Then Kv  H has a C6-decomposition if and only if Kv  H is 6-sufficient. (Ashe, Fu and Rodger, Ars Combin.) Let H be a spanning odd forest of Kv where v is even. Then Kv  H has a C6-decomposition if and only if Kv  H is 6-sufficient. (Ashe, Fu and Rodger, DM 2004)”¢5"ē’ē’ ē’ē’]ē’ē’ ē’ē’6Ŗ˜ *- %*6ó(%ŸØ!Embedding Partial 4-cycle Systems”"" Ÿ PCan we embed a partial 4-cycle system of order u in a 4-cycle system of admissible order v with v ³š u + u1/2 ? Problem : Embedding partial k-cycle systems. (Try k = 6.)”>©aˆē’€ˆ<€ó$!ŸØA do-able problem Ÿ öLet Ku  H be a partial 4-cycle system of order u where u is even and Dš(H) £š 3. Then Ku  H can be embedded in a 4-cycle system of admissible order v ³š u + u1/2. The cases when H is a 2-regular graph or a spanning odd forest have been done recently. ”ˆüē’@€ˆē’>€ˆē’€ˆˆē’W€ˆó+(ŸØPentagon SystemsŸ >Compare to 4-cycle systems or 3-cycle systems, the study of 5-cycle systems is harder. It takes a long while to find the necessary and sufficient conditions to decompose a complete 3-partite graph into C5 s. (Billington et al.) Problem: Let H be a 2-regular subgraph of Kv such that v is and odd integer, v ³š 5 and v(v-1)/2 - |E(H)| is a multiple of 5. Then Kv  H has a C5-decomposition. (Kv  H is 5-sufficient.)”¼ä¼Ė 璁$ ē’#5‚Šē’ ‚Šē’‚Šē’‚ŖbŠ & Vó,*ŸØBalanced Bipartite Designs”Ÿ |For experimental purpose, bipartite designs were introduced many years ago. Definition (BBD) A balanced bipartite design with parameter (u,v;k;lš1,lš2,lš3) (defined on X Čš Y), (X Čš Y, B), is a Kk-decomposition of lš1Ku Čš lš2Kv Čš lš3Ku,v where |X| = u and |Y| = v. Note: A pair of distinct elements from X (respectively Y) occurs together in lš1 (respectively lš2) blocks of B and two elements from different sets occur together in B exactly lš3 blocks.”.æZL ‚’3ž.‚Šē’‚Šē’‚Šē’‚ ‚‚‚ƒ ‚Šē’‚Šē’‚Šē’‚Šē’‚Šē’‚Šē’‚Šē’i‚Šē’‚Šē’ ‚‚ƒ7‚‚ƒ ‚Šē’‚Šƒē’Ŗæžó-+ŸØ An ExampleŸ ^Can we decompose the following graph into K3 s?”$0+ē’ó/-ŸØHint of SolutionŸ Š1. Let X and Y be two disjoint sets of size 5 and 11 respectively. 2. Use two vertices a and b of Y and X to define a 2K2,5. Then decompose 2K2,5 Čš K5 into K3 s. 3. Use X Čš (Y  {a,b}) to define a 2K5,9. Then use 2K5,9 and five 2-factors defined on (Y  {a,b}) to obtain a collection of K3 s. 4. Decompose the remaining part of graph defined on Y into K3 s.” iZx ē’ ē’‚Šē’‚Šē’‚ ‚Šē’‚Šē’‚(‚Šē’@‚Šē’‚Ŗfó.,ŸØPartial ResultsŸ øThe necessary conditions of the existence of a (u,v;k;lš1,lš2,lš3) BBD was transferred into several tables by Fu and Miwako Mishima for k = 3 and 4 and a few BBD s were constructed two years ago, but we are not able to finish all constructions. In case that k = 3, u = v and lš1 = lš2 we have a 3-GDD with two associates where we have two groups. Several special BBD s have been constructed by Kageyama et al. 1u¼e‰hQ萌[bś^Ėiųvvu‰Ü–, ąVdk*bóīvMR’lż€ZPŒ[ƒ[. Problem : Find all (u,v;3;lš1,lš2,lš3) BBD s.” ŻZ/‚Šē’‚Šē’‚Šē’ӂŠē’‚Šē’›‚‚‚Šē’‚Šē’‚Šē’ ‚Ŗ>r +ó0.ŸØA different approachŸ ęReplace K3 with C4, then we have a bipartite 4-cycle design denoted by (u,v;C4;lš1,lš2,lš3) BQD. (Q for quadrangle) It is quite complicate to find all BQD s, but it is possible to construct each of them. (It takes a long time to put them together.) Similar work on 4-cycle GDD with two associates was obtained earlier by Fu and Rodger. (Combin., Prob. and Computing, 2001) ”“t  ē’ ē’5‚Šē’‚Šē’‚Šē’‚Šē’‚ŖNó1/ŸØ 4-cycle GDDŖ Ÿ \ Let n, m ³š 1 and lš1, lš2 ³š 0 be integers. A 4-cycle (n,m;C4;lš1,lš2) GDD exists iff (1) 2 divides lš1(n-1) + lš2n(m-1), (2) 8 divides lš1mn(n-1) + lš2n2m(m-1), and if lš2 = 0 then 8 divides lš1n(n-1), (3) if n = 2 then lš2 > 0 and lš1 £š 2(m-1)lš2, and (4) if n = 3 then lš2 > 0 and lš1 £š 3(m-1)lš2/2 - dš(m-1)/9, where dš = 0 or 1 if lš2 is even or odd respectively. ”NŠZ%Z  ‚Šē’‚Šē’)‚Šē’‚Šē’‚Šē’,‚Šē’ ‚Šē’‚Šē’‚Šē’ ‚Šē’‚Š"‚Šē’‚Šē’5‚Šē’ ‚Šē’ ‚Šē’ ‚Šē’ ‚Šē’ ‚Šē’$‚Šē’‚‚ŖX)*Q ęó31ŸØ"Counter-part of Packing - Covering” #$ ƒ$Ÿ TAn H-covering of a graph G is a collection of its subgraphs G1, G2, & , Gt such that Gi @š H, i = 1, 2, & , t, and each edge of G is in at least one Gj for some j Īš {1,2,& ,t}. A Kk-covering of Kv is known as a k-covering of order v and the graph induced by a k-covering is a supergraph G of Kv. The graph G  Kv is known as the padding of the k-covering. A covering with minimum padding (in size) is called a minimum covering. ”(«Z= ē’ ē’ ē’  ē’‚9 ē’ ‚Šē’‚Šē’a‚Šē’‚Šē’‚‚ƒ]‚Ŗ²2   ; O tó42ŸØ Short CutŸ >If we can find an H-packing of a graph G with leave L, then P is a padding of an H-covering of G provided that L + P has an H-decomposition. ( + represents graph union.) For example, a maximum K3-packing of K11 has leave C4 and its minimum K3-covering of K11 has padding a double edge. ”Š qR€ˆē’ €ˆē’ €ˆē’€ˆē’€ˆē’€ó53ŸØMore General CoveringsŸØ’Let P be a 2-regular subgraph of Kv such that Kv + P is 3-sufficient. Then Kv + P has a K3-decomposition, i.e., P is a padding of a K3-covering of Kv. (Two groups of authors.) Let F be a spanning odd forest of Kv such that Kv + F is 3-sufficient. Then Kv + F has a K3-decomposition. (C.M. Fu, Fu and Rodger, DM) C4-covering has similar results and I believe that Ck-covering also has similar results.”“" ē’  ē’ ē’  ē’+ ē’ ē’> ē’  ē’ ē’  ē’. ē’2 ē’$Ŗ˜  F= “ó64ŸØConjectures on CoveringŸ ‚Conjecture A Let P be a subgraph of Kn such that Dš(P) £š n/4 and Kn + P is 3-sufficient. Then P is a padding of a K3-covering of Kn. Conjecture B Let P be a subgraph of Kn such that Dš(P) £š n/4 and Kn + P is 4-sufficient. Then P is a padding of a C4-covering of Kn. Note: The upper bound  n/4 is too conservative!?”0BZ ē’ ‚Šē’1‚Šē’‚Šē’‚‚‚Šē’‚Šē’0‚Šē’‚Šē’‚‚2‚Ŗt ] tó75Ÿ vn/2 is too much for the upper bound of Dš(P) in Conjecture A”0<' ‚  ŸØīExample: Let P = K3,3. Then K6 + P is 3-sufficient, but it is not K3-decomposable. (See it? There are too many bipartite edges in the graph.) 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°Š@š<$ńD 0Ć  š žšH šD ƒ š0ƒ“ŽŸ‹”޽hæ’ ?š ’’’€€€Ģ™33ĢĢĢ’²²²īAļ € ńšéšš Hš‡š( š šHšr šH S š€t\æ’š€°ŠPšĆ   š žšŸš8 š€@0p š HšP°€ š`2 šH ƒ š0…‡ƒæĄ’šą p° šf2 šH “ š6…‡‚€ƒæĄ’š€@0pšą¢ šH" c š„€š]0e‚˜²ƒ0e„˜²…†‡ˆ‰Š‹æƒæĄ’ ˆšĄ 0 ` € šĆ  š ž𙢠šH ƒ š0€\SæƒæĄ’š0 ąĄ  š1ŸØH” 2š.¢ š H ƒ š0€Ą`æƒæĄ’šą ą  šĪŸ XFor which H Kv  H has a K3-decomposition?”8- 2ē’ ē’Ŗ šB šH s š*Ģ’f“ŽŸ‹”޽hæ’ ?š ’’’€€€Ģ™33ĢĢĢ’²²²īļ€ ÉšĮ@šLšYš( š šLšX2 šL ƒ š0…‡ƒæĄ’šŠšpŠ šX2 šL ƒ š0…‡’‚€æĄ’š0PP @ šŹ¢ šL ƒ š0€Ģī鿁ƒæĄ’šŠ`@ š šjŸ Kv  H ”& 2ē’Ŗš‘¢ šL ƒ š0€Ģ 꿁ƒæĄ’šš€ Ą š1ŸØH” 2šö¢ šL ƒ š0€T"꿁ƒæĄ’š`  °€  š–Ÿ THow about this kind of H when |V(H)| »š v ?”&+ 2%€šH šL ƒ š0ƒ“ŽŸ‹”޽hæ’ ?š ’’’€€€Ģ™33ĢĢĢ’²²²ī<ļ € ģšäšPš|š( š šPšž šP S š€ÜRĘæ’š€°ŠPš<$ńD 0Ć  Ę š žšž šP S š€¼SĘæ’šą°Šš<$ńD 0Ć Ę š žšH šP ƒ š0ƒ“ŽŸ‹”޽hæ’ ?š ’’’€€€Ģ™33ĢĢĢ’²²²īRļ € šś šTš’š( š šTšž šT S š€€Ęæ’š€°ŠPš<$ńD 0Ć  Ę š žšž šT S š€`Ęæ’šą°Šš<$ńD 0Ć Ę š žš² šT C š²€€AĮšC:\Program Files\Common Files\Microsoft Shared\Clipart\cagcat50\BD00028_.WMFšą Ą ą š,$ńD  0šH šT ƒ š0ƒ“ŽŸ‹”޽hæ’ ?š ’’’€€€Ģ™33ĢĢĢ’²²²īb ļ €  š °šXš¢ š( š šXšŒ2 šX ƒ š0…‡ƒæĄ’š š€0 š,$ńD 0šž šX S š€LĘæ’š€°ŠPš<$ńD 0Ć  Ę š žšž šX S š€ZĘæ’šĄPš<$ńD 0Ć Ę š žšŒ2 šX ƒ š0…‡ƒæĄ’šp ` šš,$ńD 0šŒ2 šX ƒ š0…‡ƒæĄ’šp °š,$ńD 0šŒ2 šX ƒ š0…‡ƒæĄ’šp0 Ą š,$ńD 0šÅ š X ƒ š0€@ŽēƒæĄ’ š` Ą°  šeŸØO6m+3”.€   ē’¦ųŲŌŠššÅ š X ƒ š0€\ĘēƒæĄ’ š Ą€ą  šeŸØO6m+3”.€   ē’¦ųŲŌŠššÅ š X ƒ š0€ŒfʁƒæĄ’ šŠ ą šeŸØO6m+3”.€   ē’¦ųŲŌŠššLB š X c š$DæĄ’š@P`  šLB šX c š$DæĄ’š0 0` šLB šX c š$DæĄ’š šP Š šLB šX c š$DæĄ’šą€ą   šLB šX@ c š$DæĄ’š0` p šLB šX@ c š$DæĄ’š °° Ą šLB šX@ c š$DæĄ’š@€ ` šLB šX@ c š$DæĄ’š  š Š š¾¢ šX ƒ š0€ljĘæƒæĄ’š ą °  š^ŸØ K6m+3,6m+3”8 2’3ž  ’3žē’’3žšē¢ šX ƒ š0€ˆoĘæƒæĄ’šĄ šT  š‡ŸØGc = ”L 2$ $$ $$Ŗš¢ šX Ó šN€œuĘæƒæĄĖjJ’?æ’"ńæ‚‚šŪšĒ  š†Ÿ DG can not be decomposed into K3 s.”&#ē’šH šX ƒ š0ƒ“ŽŸ‹”޽hæ’ ?š ’’’€€€Ģ™33ĢĢĢ’²²²ī<ļ € ģšäĄštš|š( š štšž št S š€š}Ęæ’š€°ŠPš<$ńD 0Ć  Ę š žšž št S š€Š~Ęæ’šą°Šš<$ńD 0Ć Ę š žšH št ƒ š0ƒ“ŽŸ‹”޽hæ’ ?š ’’’€€€Ģ™33ĢĢĢ’²²²ī<ļ € ģšäŠš\š|š( š š\šž š\ S š€<ƒĘæ’š€°ŠPš<$ńD 0Ć  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š<…DæĄĖjJŠŃ’š0   Š0 š š!œ ć šŹ0e‚˜²ƒ0e„˜²…‡ˆ‰æ ō   BP C@DEĮFĮæĄĮĆ ÄÅAĘĮĒČÉŹĖjJĢĶĪĻĮŠŃŅÓŌÕ×’ĖĖĖ 8c8c     ?€‚ƒœ1„…šł†‡÷ˆ æĄĮĀdĆÄÅĘĒČÉŹ0uĖŠĢ0ķģ’Ķ@T‰Ī€Ļ€’’Šy’Ń2Ņ NÓPĆŌÕ'Öp”×°<’’ŲŁ'Śp”’AØ)BCD|¾E„|¾…†|¾‡š’@ō č Xˆ ąP @@­ ­ ¬€S"ń‘’æ‚‚šš€Š0 š š"œ ć šŹ0e‚˜²ƒ0e„˜²…‡ˆ‰æ ō   B€ CDEĮFĮæĄĮĆ ÄÅAĘĮĒČÉŹĖjJĢĶĪĻĮŠŃŅÓŌÕ×’ĖĖĖ 8c8c     ?€‚ƒœ1„…šł†‡÷ˆ æĄĮĀdĆÄÅĘĒČÉŹ0uĖŠĢ0ķģ’Ķ@T‰Ī€Ļ€’’Šy’Ń2Ņ NÓPĆŌÕ'Öp”×°<’’ŲŁ'Śp”’AØ)BCD|¾E„|¾…†|¾‡š’hАPŠ X€ @­ ­ ¬€S"ń‘’æ‚‚š0 €@šŖ¢ š#œ ³ šB€Ģ ÉææĄĖjJ’?æ’"ńæ‚‚š N@  š*Ÿ”(2𨢠š$œ ³ šB€ÜÉææĄĖjJ’?æ’"ńæ‚‚š; 6[  š(Ÿ”𨢠š%œ ³ šB€šÉææĄĖjJ’?æ’"ńæ‚‚š +  š(Ÿ”𿢠š&œ Ó šN€œÉ…‡ææĄĖjJ’?æ’"ńæ‚‚š: æ#Z  š3ŸØH :”šH šœ ƒ š0ƒ“ŽŸ‹”޽hæ’ ?š ’’’€€€Ģ™33ĢĢĢ’²²²ī<ļ € ģšäpšøš|š( š šøšž šø S š€T(Éæ’š€°ŠPš<$ńD 0Ć  É š žšž šø S š€4)Éæ’šą°0š<$ńD 0Ć É š žšH šø ƒ š0ƒ“ŽŸ‹”޽hæ’ ?š ’’’€€€Ģ™33ĢĢĢ’²²²īRļ € šś€š š’š( š š šž š  S š€ -Éæ’š€°ŠPš<$ńD 0Ć  É š žšž š  S š€€.Éæ’šą°Šš<$ńD 0Ć É š žš² š  C š²€€AĮšC:\Program Files\Common Files\Microsoft Shared\Clipart\cagcat50\BD05299_.WMFšP ° `5š,$ńD’’’  0šH š  ƒ 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’’’€€€Ģ™33ĢĢĢ’²²²uxœķVĶkA’ĶlR›M i,¶”‚‹oĘ`{P ‰<4Jӛ—DŻbÄd›f›²ōRؽˆųˆśčŋ‡žżļVšš›øė› ۈŗQŖł-“yo¾~ófŽ{“7Æ'÷=™ż€,Į€¤0icŗHd®u?‚°9į@į •¤¾Ć„®żæ¼§žVąŠēĀB MŖ7ą ¦‚ā’ż˜ł€æā²½«ŗĻGĒī¼·»÷·_2C(F˜S.ā&ģ”8£0ĮYŌžøó¦ņÉžÖiWpchžń‹(̉Ė/ĘÆi™kū/ŃÉŪh£ æų˜Ńö'†ą{żØ¹Æļ-Œ‘ DĪ?DeœJ āŒGyį_SĻų>œNŽķŒ<Ć0•ļ Ęž4ż,ׯn8mg͵J­Ķš[wšÖ|¾€ u«ōې&=TņóčyڊĒ/x…?' āJīC÷-š€Źc&śČ“·ęĪÉ^įé)J*Š…Ä8-!„<›Ńméžø ÆršIĒĒ&0«c.+¬&é•.-4MćÓzLĖ5÷śŖ·NI®*¹?sKĒé=ɚąČ®ÖvŪ*Ū[֊Ө5qł]oė¦iß ¦bņ”ŁCEĪģxJ?)u×Sėļųæ¶>“õ§>ĻC–—§ZōLm ķ5ŕ•¹x.Īč6ćüXÅk»vC“™śfB_Ło‹ābįSš ŹgĖĀ;žę䍎yqTĻ©ŽĀ­=•Ż Ę f²ÜnĘGš$—v‘ŪķągĖž7(”…MŌčåÆÓ;Ų¤’e­ub½Ź~t½Sdd,ĘĆņ’ndžÆ;ļ†rč oŖw–y‚{n}²ö:„~† ĄĀˆŒ¬—š™JœŽž×£ų§<Ŗ–­D¼ˆ¾˜ÉÜĖ Īŗ`ޤŌ”ŹĢĄdŠč’]īīŌ2ׂŁėØŅæč”šåņ)õ€’”żÄ°3PœąĢ`9dõ7ķG!9ž’ą…ŸņłOh«‘+'³Ł0  hpˆ˜š   , 8 DPX¶Graph-Designsohlfu-DeNC:\Program Files\Microsoft Office\Templates\Presentation Designs\Notebook.pot\hlfuogr48uMicrosoft PowerPointoso@ <Ņ ¹@ĄßsPyÄ@¾‹¼ŸƒÄĢ G@’’’’‰g  +&’’’’ĄŠ &’’’’&#’’’’TNPP2’’OMié & TNPPō &’’’’&TNPP   ŠĄ üĢ’f-ś-- !šŠĄ-ü’’’-šś-&’’’’GĮ&’’’’ūĶ«ōw@¾ q[ŌōwdŌōw0- &’’’’GyĮ&’’’’ ü--iyšH-- ūÅ’@Times New Roman[ŌōwdŌōw0-š .+2 ³Graph Decomposition vs. *+/. .'2 dÜCombinatorial Design F'.+.--Q1˜-- ūÕ’@Times New Roman[ŌōwdŌōw0-š . 2 ĘHung. . 2 Ęu-. .2 Ęƒ Lin Fu (   .ūÕ’ˆ·s²Ó©śÅéĶ }[ŌōwdŌōw0-š .2 ƳūķĄM+++.ūÕ’@Times New Roman[ŌōwdŌōw0-š . 2 Ę›).ūÕ’ˆ·s²Ó©śÅéĶ €[ŌōwdŌōw0-š .(2 õ°ź„ß„ę³q¤j¾ĒĄ³„μƾǨt+++*++*++*+.--ū¼ˆ"System 0-š&TNPP &’’’’ž’ÕĶ՜.“—+,ł®0Čˆ¤“¼ÄĢ Ō Üä ģōü  L¶ ¦pæĆ¹õ¤j¤pA-Mathj«Š2ķ 7Times New Roman ·s²Ó©śÅéSymbol ¹w³]²³ų³]­pMicrosoft ¤čµ{¦”½sæč¾¹ 3.0/Graph Decomposition vs. Combinatorial Design MotivationMy advisor”¦s comment (1995)My experiencePreliminaries Continued ”KGraph Decomposition)Balanced Incomplete Block Designs (BIBD)Another point of view More GeneralGroup Divisible DesignsGDD with two associatesGraph decomposition workst-(v,k,ƒÜ) Designs Embeddings¶i®i Continued ”KProblemPowerPoint ²³ųNecessary conditionsNash-Williams Conjecture(1970)7Example: A graph G of order 24m+12 and valency 18m+8. Known Results Continued ”K-Revised Version of Nash-Williams ConjectureAn Idea works! 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