Graph Decomposition vs. Combinatorial Design Hung-Lin Fu ( 傅恆霖 ) 國立交通大學應用數學系

Motivation The 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.

My advisor’s comment (1995) From 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 in Auburn University and I was lucky to hear the comment in a combinatorial seminar.

My experience Since 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. 我也希望是一個聰明人. 於是, 我重新再回來研 究組合設計. 但是, 這回我試著用圖的概念來協 助處理.

Preliminaries A 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.

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 K v is the graph (V,E) where E contains all the 2- element subsets of V. Hence, K v has v(v-1)/2 edges. We shall use K v to denote the complete multi- graph with multiplicity, I.e. each edge occurs times.

Graph Decomposition 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.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 we say that G has an H- decomposition denoted by H|G.If H = {H}, then we say that G has an H- decomposition denoted by H|G. An H-decomposition of K v is also known as an H- design of order v.An H-decomposition of K v is also known as an H- design of order v.

Balanced Incomplete Block Designs (BIBD) A BIBD or a 2-(v,k, ) 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 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.

Another point of view The existence of an STS(v) is equivalent to the existence of a K 3 -decomposition of K v, i.e. decomposing K v into triangles.

More General The existence of a 2-(v,k, ) design can be obtained by finding a K k -decomposition of K v. Example: 2K 4 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}.

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 n 1, n 2, …, n m, then the graph is denoted by K(n 1,n 2,…,n m ). In case that all partite sets are of the same size n, then we have a balanced complete m-partite graphs denoted by K m(n). A K k -decomposition of K m(n) is a k-GDD and a - fold k-GDD can be defined accordingly. (See it?)

GDD with two associates A group divisible design with two associates 1 and 2, GDD(n,m;k; 1, 2 ), 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 1 blocks of B and (ii) two distinct elements of X from different groups in G occur together in exactly 2 blocks of B. A k-GDD defined earlier as a K k -decomposition of K m(n) is a GDD(n,m;k;0,1). A GDD(n,m;k; 1, 2 ) can be viewed as a K k - decomposition of the union of m ( 1 K n )’s and a 2 K m(n).

Graph decomposition works Let n, m, 2  1 and 1  0. Then a GDD(n,m;3; 1, 2 ) exists if and only if (1) 2 divides 1 (n-1) + 2 (m-1)n, (2) 3 divides 1 mn(n-1) + 2 m(m-1)n 2, (3) if m = 2 then 1  2 n/2(n-1), and (4) if n = 2 then 2 (m-1)  1. (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.

t-(v,k, ) Designs Let K v (t) denote the complete t-uniform hypergraph of order v with multiplicity. Then K v (t) has edges. A t-(v,k, ) design is a K k (t) -decomposition of K v (t). A Steiner quadruple system of order v is a 3-(v,4,1) design. Note: K v is K v (2).

Embeddings An STS(u) can be embedded in STS(v) iff K v – K u has a K 3 -decomposition. A partial Steiner triple system of order u can be viewed as a subgraph H of K u. Then H can be embedded in a Steiner triple system of order v iff K v – H can be decomposed into triangles. It is conjectured that K v – 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.

進展 The conjecture has been verified for several special classes of graphs H. 1.If H is a complete graph, then it is the well-known Doyen and Wilson theorem. 2.If H is corresponding to the maximum packing of K u, then it is proved by Fu, Lindner and Rodger. 3.The version of embedding K u – H in K v does have similar results. The case when H is corresponding to the maximum packing of K u was completely settled by Su, Fu and Shen recently after an earlier effort by Milici, Quattrocchi and Shen on the case when is even.

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 K v – 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.

Problem K v – H H For which H K v – H has a K 3 -decomposition?

K v – H H How about this kind of H when |V(H)|  v ?

Necessary conditions If K v – H has a K 3 -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 K 3 -decomposition, then G is 3-sufficient.

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 K 3 -decomposition for sufficiently large n. Why 3n/4? (  (H) < n/4 where G = K n – H.)

Example: A graph G of order 24m+12 and valency 18m+8. O 6m+3 K 6m+3,6m+3 G c = G can not be decomposed into K 3 ’s.

Known Results Theorem(C. Colbourn and A. Rosa, 1986) Let H be a 2-regular subgraph of K v such that v is an odd integer not equal to 9 and v(v-1)/2 - |E(H)| is a multiple of 3. Then K v – H has a K 3 - decomposition. Note: We can also consider the above theorem as packing K v with K 3 ’s such that the leave is H. Let H = C 4  C 5. Then K 9 – H can not be decomposed into K 3 ’s. (See it?)

Continued … Theorem(Gustavsson, Ph.D. thesis 1991) Nash-Williams’ conjecture holds for the graphs which are 3-sufficient and minimum degree not less than (1 – )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. 這個問題應該有進展的空間.

Revised Version of Nash-Williams Conjecture K 3 -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 K 3 -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?

An Idea works! Adjust the leave a little bit.

Problems Let v be an even integer and H be an odd spanning forest of K v such that K v – H is 3-sufficient. Then K v – H has a K 3 - decomposition. ( 我最想解決的問題.) Let v be an even integer and H be an odd spanning subgraph of K v such that  (H) is at most 3 and K v – H is 3-sufficient. Then K v – H has a K 3 -decomposition.

Continued … Can we embed the K 3 -packings of K u 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 K u with K 3 ’s in triple systems when u is odd. We have more partial triple systems to embed now.

Cycle Systems A cycle is a connected 2-regular graph. We use C k to denote a cycle with k vertices and therefore C k has k edges. If G can be decomposed into C k ’s, then we say G has a k-cycle system and denote it by C k | G. If C k | K v, 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.

Known Results C k | K v if and only if K v is k-sufficient. Let v be even and I is a 1-factor of K v. Then C k | K v – I if and only if K v – 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))

C 3 C 4 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)

Packing with 4-cycles The maximum packing of K v with C 4 ’s has leave L i, i  Z 8 for v  i (mod 8) and L i is F, , F, C 3, F, E 6, F, C 5 depeding on i = 0, 1, 2, …, 7. Similar result as Colbourn and Rosa’s theorem: Let H be a 2-regular subgraph of K v where v is odd. Then K v – H has a C 4 -decomposition if and only if v(v-1)/2 - |E(H)| is a multiple of 4 (K v – H is 4-sufficient). (Fu and Rodger, GC 2001) Surprisingly: If H is a spanning forest of K v where v is even, then K v – H has a C 4 - decomposition iff K v – H is 4-sufficient. (Fu and Rodger, JGT 2000)

Continued … Let H be an odd graph with  (H) not greater than 3. Then K v – H has a C 4 -decomposition if and only if K v – 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 K v with  (H)  v/4 and 4  k  v. Then K v – H has a C k -decomposition if and only if K v – H is k-sufficient.  Why v/4?

An example for k = 4 K 8 – H can not be decomposed into 4-cycles. H :

Another Evidence Let H be a 2-regular subgraph of K v. Then K v – H has a C 6 -decomposition if and only if K v – H is 6-sufficient. (Ashe, Fu and Rodger, Ars Combin.) Let H be a spanning odd forest of K v where v is even. Then K v – H has a C 6 - decomposition if and only if K v – H is 6- sufficient. (Ashe, Fu and Rodger, DM 2004)

Embedding Partial 4-cycle Systems Can we embed a partial 4-cycle system of order u in a 4-cycle system of admissible order v with v  u + u 1/2 ? Problem : Embedding partial k-cycle systems. (Try k = 6.)

A do-able problem Let K u – H be a partial 4-cycle system of order u where u is even and  (H)  3. Then K u – H can be embedded in a 4-cycle system of admissible order v  u + u 1/2. The cases when H is a 2-regular graph or a spanning odd forest have been done recently.

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 C 5 ’s. (Billington et al.) Problem: Let H be a 2-regular subgraph of K v such that v is and odd integer, v  5 and v(v-1)/2 - |E(H)| is a multiple of 5. Then K v – H has a C 5 - decomposition. (K v – H is 5-sufficient.)

Balanced Bipartite Designs For experimental purpose, bipartite designs were introduced many years ago. Definition (BBD) A balanced bipartite design with parameter (u,v;k; 1, 2, 3 ) (defined on X  Y), (X  Y, B), is a K k -decomposition of 1 K u  2 K v  3 K u,v where |X| = u and |Y| = v. Note: A pair of distinct elements from X (respectively Y) occurs together in 1 (respectively 2 ) blocks of B and two elements from different sets occur together in B exactly 3 blocks.

An Example Can we decompose the following graph into K 3 ’s? K5K5 3K 11 2K 5,11 Quiz : Does a (5,11;3;1,3,2) BBD exist?

Hint of Solution 1. 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 2K 2,5. Then decompose 2K 2,5  K 5 into K 3 ’s. 3. Use X  (Y – {a,b}) to define a 2K 5,9. Then use 2K 5,9 and five 2-factors defined on (Y – {a,b}) to obtain a collection of K 3 ’s. 4. Decompose the remaining part of graph defined on Y into K 3 ’s.

Partial Results The necessary conditions of the existence of a (u,v;k; 1, 2, 3 ) 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 1 = 2 we have a 3- GDD with two associates where we have two groups. Several special BBD’s have been constructed by Kageyama et al. 由於要全部完成建構相當複雜, 因此截至目前沒 能做完它. Problem : Find all (u,v;3; 1, 2, 3 ) BBD’s.

A different approach Replace K 3 with C 4, then we have a bipartite 4- cycle design denoted by (u,v;C 4 ; 1, 2, 3 ) 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)

4-cycle GDD Let n, m  1 and 1, 2  0 be integers. A 4-cycle (n,m;C 4 ; 1, 2 ) GDD exists iff (1) 2 divides 1 (n-1) + 2 n(m-1), (2) 8 divides 1 mn(n-1) + 2 n 2 m(m-1), and if 2 = 0 then 8 divides 1 n(n-1), (3) if n = 2 then 2 > 0 and 1  2(m-1) 2, and (4) if n = 3 then 2 > 0 and 1  3(m-1) 2 /2 -  (m-1)/9, where  = 0 or 1 if 2 is even or odd respectively.

Counter-part of Packing - Covering An H-covering of a graph G is a collection of its subgraphs G 1, G 2, …, G t such that G i  H, i = 1, 2, …, t, and each edge of G is in at least one G j for some j  {1,2,…,t}. A K k -covering of K v is known as a k-covering of order v and the graph induced by a k-covering is a supergraph G of K v. The graph G – K v is known as the padding of the k-covering. A covering with minimum padding (in size) is called a minimum covering.

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 K 3 -packing of K 11 has leave C 4 and its minimum K 3 -covering of K 11 has padding a double edge.

More General Coverings Let P be a 2-regular subgraph of K v such that K v + P is 3-sufficient. Then K v + P has a K 3 - decomposition, i.e., P is a padding of a K 3 - covering of K v. (Two groups of authors.) Let F be a spanning odd forest of K v such that K v + F is 3-sufficient. Then K v + F has a K 3 - decomposition. (C.M. Fu, Fu and Rodger, DM) C 4 -covering has similar results and I believe that C k -covering also has similar results.

Conjectures on Covering Conjecture A Let P be a subgraph of K n such that  (P)  n/4 and K n + P is 3-sufficient. Then P is a padding of a K 3 -covering of K n. Conjecture B Let P be a subgraph of K n such that  (P)  n/4 and K n + P is 4-sufficient. Then P is a padding of a C 4 - covering of K n. Note: The upper bound “n/4” is too conservative!?

n/2 is too much for the upper bound of  (P) in Conjecture A Example: Let P = K 3,3. Then K 6 + P is 3- sufficient, but it is not K 3 -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.

感謝的話 很高興有機會在靜宜演講. 黃教授在百忙中接辦這個研討會, 讓剛畢 業的新苗有機會告訴大家做了哪些新工 做, 在此替我的學生致上最大謝意. 祝研 討會順利成功.