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TV    )'   !%+,-./01234o2$uU(0zg5BGV%2$aNh`B%$2$*@źgBqW*.&$2$}|n1 B 0e0e    A AjJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||S"@3fff3@fm`dʚ;2Nʚ;g4VdVdЬ 0@ppp@ <4!d!d &) 0Ts<4dddd &) 0Ts .___PPT10De0}fԚ New RomanTTx 0xDTimes New RomanTTx 0x ppN___PPT90(E? %R4-cycle Designs6<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.  LFFrom Graphs to Designs7Since 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. Graph decomposition is a great tool in constructing beautiful designs! 8H3- 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. We shall use lG to denote the multi-graph obtained from G with multiplicity l, i.e. each edge of G occurs l times.X_Z> Graph Decomposition We 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 G has an H-decomposition or an H-design.  .e%)3  A Famous Example ! (1847)zThe existence of an STS(v) is equivalent to the existence of a K3-decomposition of Kv, i.e. decomposing Kv into triangles.H{@,S Cycle DesignsA 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 design 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, STS(v).aZ1 2 + 9    ,}Necessary conditions@If G has a C3-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 G has at least x vertices, x | |E(G)| and G is an even graph. If G has a decomposition into k-cycles, then G must be k-sufficient !FAZ m 3wF;94-cycle Designs_A 4-cycle design of the complete graph of order v is also known as a 4-cycle system of order v.`\$ TExample : v = 9. (0,1,5,3), (1,2,6,4), & ./4-cycle Designs of Complete Multipartite Graphs00A 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).    "   x 4}:8A Beautiful 4-cycle DesignA 4-cycle design of the complete multipartite graph G exists if and only if G is 4-sufficient. In fact, finding the maximum packing with maximum number of 4-cycles in the complete multipartite graph is also possible. (Billington, Fu and Rodger, JCD 9)@Z$ $3$$ MGOther Nice 4-cycle DesignsA cyclic 4-cycle design of the complete graph of order v exists if and only if v a" 1(mod 8). (Well-known) A resolvable 1-rotational 4-cycle design of 2Kv exists if and only if v a" 0 (mod 4). (Fu and Mishima, 2002 JCD) An almost resolvable directed 4-cycle design of Dv exists if and only if v a" 1(mod 4). (Fu, Fu, Milici, Quattrochi and Rodger, 1995 JCD) dI""&"*""&3"&3&&3&&3"&3"*"" "0&3&3"X</ NHContinued & Group Divisible Designs 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?) 0Z   "   x   (ZGOI GDD with two associates A 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). Z- ? "  ## H~=9RJ 4-cycle GDDLet 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. (Fu and Rodger, Combin., Prob. and Computing, 2001) tZ]ZZ  !!  "5     $"136MZ"SM ~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 $PL"Balanced Bipartite 4-cycle Designs##(Replace 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.) Balanced bipartite 4-cycle designs does exist. (Australasian J. Combin. 2005) H   536 QK  ProblemKv  H . 2 KEKnown H s"H is a complete graph of order not too close to v. H is a disjoint union of 4-cycles such that the order of H is not too close to v. (*) If H is of order u, then v has to be at least u + u1/2. So far, no one can prove this conjecture. The best known result is that v is at least 2u + 15. l" Z" Z" Z3 c%"|E(H)| is small!3 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) 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)K# EO QN1&#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 (Chen and Fu) Let H be a subgraph of Kv with D(H) < [(v-1)/3]. Then Kv  H has a C4 decomposition if and only if Kv  H is k-sufficient. Why [(v-1)/3]?ZZ  d 3     333t:+$+*'$ An example for k = 4XK8  H can not be decomposed into 4-cycles. ,-+JDGraphs with no C4-decomposition6 (3(3(34K3(2k+1) + C5 where k 3.@ <:How many 4-cycles in a graph?If we have a 4-cycle design of G, then we have a bunch of 4-cycles in G. Even if G does not have a 4-cycle design, G many have some 4-cycles in there. So, it is interesting to know for which G with size e(G) as large as possible and G contains no 4-cycles !l><Tougher Case : 4-cycleLet ex(n,H) denote the size of an extremal graph with forbidden graph H. If H is a 3-cycle, then ex(n,H) is about n2/4. If H = Kr,r, then ex(n,H) is about cn 2  1/r where c is a suitable constant (who knows?) Now, how about r = 2?r(   (  (     ,"qK?=Partial ResultsLet q be a prime power. Then ex(n,C4) is equal to n(1 + (4n-3)1/2)/4 where n = q2 + q + 1. (From the existence of a projective plane of order q.) 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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|~~~~~~~|||~~~~~~~~||~~~~~~~~ dZ!    )'   !+,-./012345678o2$uU(0zg5BGV%2$aNh`B%$2$*@źgBqW*.&$2$}|n1 B 0e0e    A AjJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||S"@3fff3@fm`dʚ;2Nʚ;g4GdGdЬ 0ppp@ <4!d!d&) 0Ps<4dddd&) 0Ps .___PPT10De0}fԚ New RomanPPt 0tDTimes New RomanPPt 0t ppN___PPT90(E? %\4-cycle Designs6<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.  LFFrom Graphs to Designs7Since 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. Graph decomposition is a great tool in constructing beautiful designs! 8H3- 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. We shall use lG to denote the multi-graph obtained from G with multiplicity l, i.e. each edge of G occurs l times.X_Z> Graph Decomposition We 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 G has an H-decomposition or an H-design.  .e%)3  A Famous Example ! (1847)zThe existence of an STS(v) is equivalent to the existence of a K3-decomposition of Kv, i.e. decomposing Kv into triangles.H{@,S Cycle DesignsA 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 design 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, STS(v).aZ1 2 + 9    ,}Necessary conditions@If G has a C3-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 G has at least x vertices, x | |E(G)| and G is an even graph. If G has a decomposition into k-cycles, then G must be k-sufficient !FAZ m 3wF;94-cycle Designs_A 4-cycle design of the complete graph of order v is also known as a 4-cycle system of order v.`\$ TExample : v = 9. (0,1,5,3), (1,2,6,4), & ./4-cycle Designs of Complete Multipartite Graphs00A 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).    "   x 4}:8A Beautiful 4-cycle DesignA 4-cycle design of the complete multipartite graph G exists if and only if G is 4-sufficient. In fact, finding the maximum packing with maximum number of 4-cycles in the complete multipartite graph is also possible. (Billington, Fu and Rodger, JCD 9)@Z$ $3$$ MGOther Nice 4-cycle DesignsA cyclic 4-cycle design of the complete graph of order v exists if and only if v a" 1(mod 8). (Well-known) A resolvable 1-rotational 4-cycle design of 2Kv exists if and only if v a" 0 (mod 4). (Fu and Mishima, 2002 JCD) An almost resolvable directed 4-cycle design of Dv exists if and only if v a" 1(mod 4). (Fu, Fu, Milici, Quattrochi and Rodger, 1995 JCD) dI""&"*""&3"&3&&3&&3"&3"*"" "0&3&3"X</ NHContinued & Group Divisible Designs 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?) 0Z   "   x   (ZGOI GDD with two associates A 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). Z- ? "  ## H~=9RJ 4-cycle GDDLet 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. (Fu and Rodger, Combin., Prob. and Computing, 2001) tZ]ZZ  !!  "5     $"136MZ"SM ~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 $PL"Balanced Bipartite 4-cycle Designs##(Replace K3 with C4, then we have a bipartite 4-cycle design denoted by (u,v;C4;l1,l2,l3) BQD. (Q for quadrangle) It is quite complicate to find all BQD s, but it is possible to construct each of them. (It takes a long time to put them together.) Balanced bipartite 4-cycle designs does exist. (Australasian J. Combin. 2005) H   536 QK  ProblemKv  H . 2 KEKnown H s"H is a complete graph of order not too close to v. H is a disjoint union of 4-cycles such that the order of H is not too close to v. (*) If H is of order u, then v has to be at least u + u1/2. So far, no one can prove this conjecture. The best known result is that v is at least 2u + 15. l" Z" Z" Z3 c%"|E(H)| is small!3 Let H be a 2-regular subgraph of Kv where v is odd. Then Kv  H has a C4-decomposition if and only if v(v-1)/2 - |E(H)| is a multiple of 4 (Kv  H is 4-sufficient). (Fu and Rodger, GC 2001) If H is a spanning forest of Kv where v is even, then Kv  H has a C4-decomposition iff Kv  H is 4-sufficient. (Fu and Rodger, JGT 2000)K# EO QN1&#Continued & Let H be an odd graph with D(H) not greater than 3. Then Kv  H has a C4-decomposition if and only if Kv  H is 4-sufficient except two special cases when v = 8. (C.M. Fu, Fu, Rodger and Smith, DM 2004) Conjecture (Chen and Fu) Let H be a subgraph of Kv with D(H) < [(v-1)/3]. Then Kv  H has a C4 decomposition if and only if Kv  H is k-sufficient. Why [(v-1)/3]?ZZ  d 3     333t:+$+*'$ An example for k = 4XK8  H can not be decomposed into 4-cycles. ,-+JDGraphs with no C4-decomposition6 (3(3(34K3(2k+1) + C5 where k 3.@ <:How many 4-cycles in a graph?If we have a 4-cycle design of G, then we have a bunch of 4-cycles in G. Even if G does not have a 4-cycle design, G many have some 4-cycles in there. So, it is interesting to know for which G with size e(G) as large as possible and G contains no 4-cycles !l><Tougher Case : 4-cycleLet ex(n,H) denote the size of an extremal graph with forbidden graph H. If H is a 3-cycle, then ex(n,H) is about n2/4. If H = Kr,r, then ex(n,H) is about cn 2  1/r where c is a suitable constant (who knows?) Now, how about r = 2?r(   (  (     ,"qK?=Partial ResultsLet q be a prime power. Then ex(n,C4) is equal to n(1 + (4n-3)1/2)/4 where n = q2 + q + 1. (From the existence of a projective plane of order q.) How about the cases n is not of this form?R$A,TNRemarkstEven we have enough number of edges to guarantee that there is a 4-cycle in the graph, we still don t know if the graph can be decomposed into 4-cycles. But, it is a starting point! The focus of decomposition problem of graphs is on general graphs with certain constraints. (Not decomposing complete graphs here.)0;3UO More Remarks 1. On minimum degree depending on H: fH(n), lim fH(n)/n is a constant less than 1. 2. On minimum degree of H, d(H). We would expect that d(H) is larger, say 2. So, how about H is a 4-cycle? We list three results here (in next slides) for references.,3 33>>+VP ReferencesZR. Yuster, Tree packing of graphs, Random Struct. Algorith., 12 (1998), 237  251. R. Yuster, Decomposing large graphs with small graphs of high density, JGT 32 (1999), 27  40. T. Gustavsson, Decompositions of large graphs and digraphs with high minimum degree, Ph. D. Thesis, Univ. Stockholm, 1991. ..b!Y oWQ 0I ll stop here! Thanks.2PPP3$  0 PX$(  Xr X S