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 1994-1995 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
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}.
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?)
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).
Embeddings
• An 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.
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
Ku, then it is proved by Fu, Lindner and Rodger.
3. The version of embedding Ku – H in Kv does
have similar results. The case when H is
corresponding to the maximum packing of Ku 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 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.
Problem
H
Kv – H
For which H Kv – H has a K3-decomposition?
Kv – H
H
How about this kind of H when |V(H)|  v ?
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.
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?
((H) < n/4 where
G = Kn – H.)
Example: A graph G of order
24m+12 and valency 18m+8.
O6m+3
O6m+3
K6m+3,6m+3
Gc =
O6m+3
O6m+3
G can not be
decomposed
into K3’s.
Known Results
• Theorem(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 K3decomposition.
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?)
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 – 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. 這個問題應該有進展的空間.
Revised Version of
Nash-Williams Conjecture
• K3-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 5cycle 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 Kv such that Kv – H is
3-sufficient. Then Kv – H has a K3decomposition. (我最想解決的問題.)
• Let v be an even integer and H be an odd
spanning subgraph of Kv such that (H) is at
most 3 and Kv – H is 3-sufficient. Then
Kv – H has a K3-decomposition.
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.
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))
C3
C4
• 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 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 C4decomposition iff Kv – H is 4-sufficient. (Fu and
Rodger, JGT 2000)
Continued …
• Let H be an odd graph with (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 (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?
An example for k = 4
K8 – H can not be decomposed into 4-cycles.
H:
Another Evidence
• Let 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 C6decomposition if and only if Kv – H is 6sufficient. (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 + u1/2 ?
• Problem : Embedding partial k-cycle
systems. (Try k = 6.)
A do-able problem
• Let Ku – H be a partial 4-cycle system of
order u where u is even and (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.
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 3partite 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.)
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.
An Example
• Can we decompose the following graph into
K3’s?
K5
2K5,11
3K11
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
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.
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 3GDD 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 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.)
• 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.
Counter-part of Packing - Covering
• An 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.
Short Cut
• If we can find an H-packing of a graph G
with leave L, then P is a padding of an Hcovering of G provided that L + P has an Hdecomposition. (“+” 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.
More General Coverings
• Let P be a 2-regular subgraph of Kv such that Kv +
P is 3-sufficient. Then Kv + P has a K3decomposition, i.e., P is a padding of a K3covering 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 K3decomposition. (C.M. Fu, Fu and Rodger, DM)
• C4-covering has similar results and I believe that
Ck-covering also has similar results.
Conjectures on Covering
Conjecture A
Let P be a subgraph of Kn such that (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 (P)  n/4 and
Kn + P is 4-sufficient. Then P is a padding of a C4covering of Kn.
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 = K3,3. Then K6 + P is 3sufficient, 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.
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.