Graph Decomposition and its Applications 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). What else can we apply this wonderful outcome?

C. C. Lindner’s comment Many smart combinatorists who devoted themselves to be “graph theorists”, 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.)

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. Then, everything is Decomposition! After I know Group Testing, I have more confidence to say: Decomposition is great!

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}.

Pairwise Balanced Designs If K v can be decomposed into complete subgraphs of order in a prescribed set K, then we have a 2-(v,K, ) design, also known as a (v,K, ) 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!

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?)

k-GDD with Specified Types If the group size of a GDD is replaced with groups of different sizes t 1, t 2, …, t m, then we have a k-GDD with type t 1 × t 2 × …× t m. The GDD defined on K m(n) is of type n m. A GDD of type n m is an (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.)

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).

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))

4-Cycle Systems A 4-cycle system of order v exists if and only if v  1 (mod 8). (Use Alex Rosa’s idea.) A mapping  from V(G) into {0, 1, 2, …, |E(G)|} is an  -labeling if {|  (u) -  (v)| : uv is an edge of G} = {1, 2, 3, …, |E(G)|} and there exists a such that for each uv in E(G), either  (u)  <  (v) or  (v)  <  (u). C 4 has an  -labeling. (See it?) So are the cycles of length 4k.(Exercise!) A labeling without the second condition is called a  -labeling or a graceful labeling.

A Beautiful Idea! Theorem (Alex Rosa, 1966) If a graph G of size q has an  -labeling, then K 2q+1 can be decomposed into copies of G. Proof. Use difference method! Theorem (A. Rosa) If a graph G of size q has an  -labeling, then K 2pq+1 can be decomposed into copies of G. Proof. Now, we have p starters.

More 4-Cycle Systems 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).

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.) (*) It is done for C 3, C 4 and C 6.

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.

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.) AJC, 2005 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.

Applications Experimental Designs Group Testings DNA library Screening Scheduling Sharing Scheme Synchronous Optical Networks More …

d-Disjunct Matrices Theorem(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 M r 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 …

Group Testing Theorem (Fu and Hwang) For each r < t, M r 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.

Library Screening 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!)

Continued … 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. 長話短說 … The problem turns out to be decomposing K n into K r × K c ‘s. (Or good packings!)

Main Results K 2 × K 3 case was settled by J. E. Carter (1989). K 3 × K 3 case by Fu et al. J.S.P.I. (2003). K 2 × K 4 case by Fu et al. SIAM J. Discrete Math. (2003). What’s next?

Scheduling via Edge-Coloring 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.)

Sharing Scheme via Latin Square The existence of a latin square of order n is equivalent a decomposition of K n,n,n into triangles. Here each partite set of K n,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 n 2 /4. (Open) (Su Do Ku!) Split the entries of a critical set nicely creats a sharing scheme.

Synchronous Optical Networks 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 K N into subgraphs of size at most C with the total number of orders of subgraphs a “Minimum”.

C = 4 N = 9 : A 4-cycle system of order 9 works. An H-design of order 9 where H is K 4 – P 3 also works. (How?) (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?

The object Decomposing the complete graph of order N into as many subgraphs H with max. ratio  =  (H) /  (H) as possible! (  (H)  C.) For example, C = 6. Choose K 4. (Almost done by Bermond et al. SIAM D.M.) C = 7, K 4 works well. (Why?) C = 8. K 5 - e - f. Note: Not necessarily be maximum packings.

More … It is your term to find them out, good luck to you and all of us. Thank you for your patience! 圖分割圖分割 作業