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30103RIFFWAVEfmt ++data~~~~~~~~~~~~~~~~~~~~~~~~~~~|||~~~~~zvtvxz|~zvrnlrv||vtrpptz~|xvtv|~~zxvvvz|~xrlhntzzvrrpprrx~|j[QU_|bICCYn[ICY~zlnh]_r|]SUSSjz|x__l~v_]drnUb|nfd_]d]5=rrnj]jz~lMldrx[[_f|YQfWK_xh[zx[CIjzxxdSQz|_fI9WӹM;QnvK;OUx~xj]]YS~ɵlM?plM;CGpٵ[)Kh|% %;xtKQS]ɖbGr|lSnvz~nfS[nëx;Az=+AYݻ|=/1r_ ?|潄W5/Czãf)/pvbMQr|O=OtjdhjlG?G_ɊK3לdQCMppW;1SŷAMz/#?bx͖SAM[hvhYp~~zrvt[]ptWOUtxxd]v~vvxzd_xzh_nrSQltbdp~_lrQ_l][tf]W]~hWfp|fWhnhbU[hzlSCWbYbnxpdhjvjM9Ot]GQ[dpp~hhhdpx~xh]Yntnrp]dlvjQMb|vdjpzxzztljr||_[r~~jb_j|z|xpp~~v]Wh||zx|x|zzndhpxr]Yfntzjl|z||~vppnrzphrxz|~|xtrjjtv~~vz|z||xrrv~xrprtz|||vzxvx|xtx~~~~~~~zxxxx|zxz~~~rrz~|xz~~|vtx|zxz|~~~||zx|||~~zz|zz~~~~|~~||~~||zz|~||z~~~~zxzz|~~~~|~~~|~||~~~~~~~~~~~~~|||~~~~~~~~~~~~~||~~||~||~~~~|||~~~~||||~~~~~~||||~~~~~~~~~~~|~~~~|zz~~~~~~~||||~~||~~~~~|||~~~~~|||~|~|~~~~|z|~~~~~~~|||~~~~~~~~||~~~~~~~~ @3     !  ")$%&'(# '*+-,./01234 _2$.ĕrtVk)2$aNh`B)2$m:l8MT>?>=2$0/6gΛ&?2$s|"$̝ئ-`LA 0e0e     AjJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||S"0 3fff@uʚ;2Nʚ;g4CdCd0pppp@ <4!d!d 0,x4<4dddd 0,x4 f^___PPT9@8  ? %!.Graph Decomposition vs. Combinatorial Design <Hung-Lin Fu (PF`) WzN'Yx[a(uxex[| Motivation6The 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. 6My advisor s comment (1995)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.Z,;L My experienceHSince 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 PpfN. e/f, b͑eQVOxvzD}T-. FO/f, VbfW(uWvi_OTSRUt. %%,- 2 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%V )+* (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}. ^( ,AGroup Divisible DesignsA 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   (,} ]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    *  $ ,P $"t-(v,k,l) Designs Let lKv(t) denote the complete t-uniform hypergraph of order v with multiplicity l. Then lKv(t) has l edges. A t-(v,k,l) design is a Kk(t)-decomposition of lKv(t). A Steiner quadruple system of order v is a 3-(v,4,1) design. Note: Kv is Kv(2). ZZG ,I" '.K  EmbeddingsTAn STS(u) can be embedded in STS(v) iff Kv  Ku has a K3-decomposition. A partial Steiner triple system of order u can be viewed as a subgraph H of Ku. Then H can be embedded in a Steiner triple system of order v iff Kv  H can be decomposed into triangles. It is conjectured that Kv  H can be decomposed into triangles if v > 2u and v 1 or 3 (mod 6). Note: H is an even graph with 3t edges for some non-negative integer. Z)   ] D ? 6CP$[ G= 2U\ 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 lKu  H in lKv does have similar results. The case when H is corresponding to the maximum packing of lKu was completely settled by Su, Fu and Shen recently after an earlier effort by Milici, Quattrocchi and Shen on the case when l is even.JZ" ZJ H Yb$ Continued & 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  , ytProblemKv  H . 2 Necessary conditions<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 ZZ m 3h-Nash-Williams Conjecture(1970) 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? (D(H) < n/4 where G = Kn  H.)lZq* 33333"6Example: A graph G of order 24m+12 and valency 18m+8. 'O6m+3& Known ResultsTheorem(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 C5. Then K9  H can not be decomposed into K3 s. (See it?)^'ZUZF ^   5    !b - ]XrContinued & 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.  POULar g2U\vzz.Z'ZZo$  ,Revised Version of Nash-Williams ConjectureK3-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? ZZ  y  CwR !An Idea works!Adjust the leave a little bit.ProblemstLet 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. (bg`zlvOUL.) Let v be an even integer and H be an odd spanning subgraph of Kv such that D(H) is at most 3 and Kv  H is 3-sufficient. Then Kv  H has a K3-decomposition.;= &  A  < % 2 Continued & ,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. dV>  r d 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 )&C3 C46<A 4-cycle system of order v exists if and only if v 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)$4 %"Packing with 4-cycles*The maximum packing of Kv with C4 s has leave Li, i Z8 for v i (mod 8) and Li is F, , F, C3, F, E6, F, C5 depeding on i = 0, 1, 2, & , 7. Similar result as Colbourn and Rosa s theorem: Let H be a 2-regular subgraph of Kv where v is odd. Then Kv  H has a C4-decomposition if and only if v(v-1)/2 - |E(H)| is a multiple of 4 (Kv  H is 4-sufficient). (Fu and Rodger, GC 2001) Surprisingly: If H is a spanning forest of Kv where v is even, then Kv  H has a C4-decomposition iff Kv  H is 4-sufficient. (Fu and Rodger, JGT 2000)Z    q E? V ( ) O[0&#Continued & Let H be an odd graph with D(H) not greater than 3. Then Kv  H has a C4-decomposition if and only if Kv  H is 4-sufficient except two special cases when v = 8. (C.M. Fu, Fu, Rodger and Smith, DM 2004) Conjecture(Fu) Let H be a subgraph of Kv with D(H) v/4 and 4 k v. Then Kv  H has a Ck-decomposition if and only if Kv  H is k-sufficient. Why v/4?ZZ  d *'  3t:* %*#'$ An example for k = 4XK8  H can not be decomposed into 4-cycles. ,-++*'Another EvidencehLet 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" ] 6 *- %*6(%!Embedding Partial 4-cycle Systems"" PCan we embed a partial 4-cycle system of order u in a 4-cycle system of admissible order v with v u + u1/2 ? Problem : Embedding partial k-cycle systems. (Try k = 6.)>a<$!A do-able problem Let Ku  H be a partial 4-cycle system of order u where u is even and D(H) 3. Then Ku  H can be embedded in a 4-cycle system of admissible order v u + u1/2. The cases when H is a 2-regular graph or a spanning odd forest have been done recently. @>W+(Pentagon Systems>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 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.) $ #5 b & V,*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..ZL 3.  i 7 -+ An Example^Can we decompose the following graph into K3 s?$0+/-Hint of Solution1. 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 K5 into K3 s. 3. Use X (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   (@f.,Partial ResultsThe necessary conditions of the existence of a (u,v;k;l1,l2,l3) 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 l1 = l2 we have a 3-GDD with two associates where we have two groups. Several special BBD s have been constructed by Kageyama et al. 1uehQ萌[b^ivvuܖ, Vdk*bvMRlZP[[. Problem : Find all (u,v;3;l1,l2,l3) BBD s. Z/ >r +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.) Similar work on 4-cycle GDD with two associates was obtained earlier by Fu and Rodger. (Combin., Prob. and Computing, 2001) t   5N1/ 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     $X)*Q 31"Counter-part of Packing - Covering #$ $TAn H-covering of a graph G is a collection of its subgraphs G1, G2, & , Gt such that Gi @ H, i = 1, 2, & , t, and each edge of G is in at least one Gj for some j {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=     9  a]2   ; O t42 Short Cut>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.  qR  53More General CoveringsLet 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."      +  >      . 2 $  F= 64Conjectures on CoveringConjecture A Let P be a subgraph of Kn such that D(P) 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 D(P) 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  102t ] t75vn/2 is too much for the upper bound of D(P) in Conjecture A0<'   Example: 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.) Remark : I have a feeling that Conjecture B can be verified in the near future, may be by you.f %JW97avq_ؚ g_jg(W\[o. ÞYec(W~v_-Nc Pxg, [Rbumiveׂ g_jgJT4'Y[ZPNTNe]ZP, (Wdkfbvx[u Ng'Ya. ]yxg)RbR. 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Known Results Continued K-Revised Version of Nash-Williams ConjectureAn Idea works! Problems Continued KCycle SystemsKnown ResultsC3 C4Packing with 4-cycles Continued K An example for k = 4Another Evidence"Embedding Partial 4-cycle SystemsA do-able problem Pentagon SystemsBalanced Bipartite Designs An ExampleHint of SolutionPartial ResultsA different approach 4-cycle GDD#Counter-part of Packing - Covering Short CutMore General CoveringsConjectures on Covering=n/2 is too much for the upper bound of (P) in Conjecture A Pª  ϥΦr ²]pdO OLE A{ vD2_7"hlfuhlfu  !"#$%&'()*+,-./123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ACDEFGHIKLMNOPQSTUVWXY^Root EntrydO)PicturesP_Current UserRSummaryInformation(BPowerPoint Document(0["DocumentSummaryInformation8J