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
Kv is the graph (V,E) where E contains all the 2element subsets of V. Hence, Kv has v(v-1)/2
edges.
• We shall use Kv to denote the complete multigraph with multiplicity  , I.e. each edge occurs 
times.
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
Hdecomposition.
• If H = {H}, then we say that G has an Hdecomposition denoted by H|G.
• An H-decomposition of Kv is also known as an Hdesign 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 kelement 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 K3-decomposition of Kv,
i.e. decomposing Kv into triangles.
More General
• The existence of a 2-(v,k,) design can be
obtained by finding a Kk-decomposition of
Kv.
• 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}.
Pairwise Balanced Designs
• If Kv 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 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 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 t1, t2, …, tm, then
we have a k-GDD with type t1× t2× …× tm.
• The GDD defined on Km(n) is of type nm.
• A GDD of type nm 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 Kk-decomposition of
Km(n) is a GDD(n,m;k;0,1).
• A GDD(n,m;k;1,2) can be viewed as a Kkdecomposition of the union of m (1Kn)’s and a
2Km(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 1mn(n-1) + 2m(m-1)n2,
(3) if m = 2 then 1  2n/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 Kv(t) denote the complete t-uniform
hypergraph of order v with multiplicity .
v
(t)
Then Kv has Ct edges.
• A t-(v,k,) design is a Kk(t)-decomposition
of Kv(t).
• A Steiner quadruple system of order v is a
3-(v,4,1) design.
Note: Kv is Kv(2).
Cycle Systems
• A 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.
Known Results
• Ck | 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))
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).
• C4 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
K2q+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
K2pq+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 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 C5decomposition. (Kv – H is 5-sufficient.)
(*) It is done for C3, C4 and C6.
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 Kk-decomposition of 1Ku 
2Kv  3Ku,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 K3 with C4, then we have a bipartite 4cycle design denoted by (u,v;C4;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;C4;1,2) GDD exists iff
(1) 2 divides 1(n-1) + 2n(m-1),
(2) 8 divides 1mn(n-1) + 2n2m(m-1), and
if 2 = 0 then 8 divides 1n(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
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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 Mr 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, Mr is a d-disjunct matrix with
 k   t  1
  1.
d    / 
 r   r 
k

r

 
(*) 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
Kn into Kr × Kc ‘s. (Or good packings!)
Main Results
• K2 × K3 case was settled by J. E. Carter (1989).
• K3 × K3 case by Fu et al. J.S.P.I. (2003).
• K2 × K4 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 Kn,n,n into triangles.
Here each partite set of Kn,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 n2/4. (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 KN 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 K4 – P3
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 K4. (Almost
done by Bermond et al. SIAM D.M.)
• C = 7, K4 works well. (Why?)
• C = 8. K5 - 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!
作業
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