Triple Systems from
Graph Decompositions
Robert “Bob” Gardner
Department of Mathematics
East Tennessee State University
2008 Fall Southeastern Meeting
American Mathematical Society
Special Session on Graph Decompositions
University of Alabama, Huntsville
October 25, 2008
1. Decompositions
Definition. A decomposition of a simple
graph H with isomorphic copies of
graph G is a set
{ G1, G2, … , Gn}
where Gi  G and V(Gi)  V(H) for all i,
E(Gi) ∩ E(Gj) = Ø if i ≠ j, and
n
 Gi = H.
i 1
Note. Decompositions of digraphs and mixed graphs are
siimilarly defined.
Example. There is a decomposition of
K5 into 5-cycles.
Example. There is a decomposition of K7 into
3-cycles:
0
(0,1,3)
(1,2,4)
1
6
2
5
4
3
(2,3,5)
(3,4,6)
(4,5,0)
(5,6,1)
(6,0,2)
Definition. We shall restrict today’s presentation to
decompositions of complete graphs (or complete
digraphs or complete mixed graphs) into
isomorphic copies of graphs on 3 (non-isolated)
vertices. We refer to any such decomposition as a
triple system.
2. Steiner Triple
Systems
Definition. A Steiner triple system of order
v, STS(v), is a decomposition of the complete
graph on v vertices, Kv , into 3-cycles.
Jakob Steiner
1796-1863
v ≡ 1 or 3 (mod 6) is necessary.
From the Saint Andrews MacTutor History
of Mathematics website.
J. Steiner, Combinatorische Aufgabe, Journal für die
Reine und angewandte Mathematik (Crelle’s Journal),
45 (1853), 181-182.
Theorem. A STS(v) exists if and only if
v ≡ 1 or 3 (mod 6).
Note. Sufficiency follows from Reiss.
M. Reiss, Über eine Steinersche combinatorsche
Aufgabe welche in 45sten Bande dieses Journals,
Seite 181, gestellt worden ist, Journal für die
Reine und angewandte Mathematik (Crelle’s
Journal), 56 (1859), 326-344.
Thomas P. Kirkman
1806-1895
STS(v) iff v ≡ 1 or 3 (mod 6).
From the Saint Andrews MacTutor History of
Mathematics website.
T. Kirkman, On a problem in combinations, Cambridge
and Dublin Mathematics Journal, 2 (1847), 191-204.
Constructions Based on Difference Methods
Heffter posed two difference problems:
L. Heffter, Ueber Triplesysteme, Math. Ann., 49 (1897),
101-112.
The Problems were solved by Peltesohn:
R. Peltesohn, Eine Lösung der beiden Heffterschen
Differenzenprobleme, Compositio Math., 6 (1939), 251257.
Theorem. A STS(v) admitting a cyclic
automorphism exists if and only if
v ≡ 1 or 3 (mod 6), v ≠ 9.
3. Mendelsohn and
Directed Triple
Systems
Note. There are two orientations of a
3-cycle:
3-circuit
Transitive Triple
Nathan S. Mendelsohn
1917-2006
From the Saint Andrews MacTutor
History of Mathematics website.
N. S. Mendelsohn, A Natural Generalization of Steiner
Triple Systems, in: Computers in Number Theory
(Academic Press, New York, 1971), 323-338.
Theorem. A Mendelsohn triple system
of order v exists if and only if v ≡ 0 or 1
(mod 3), v ≠ 6.
Theorem. A directed triple system of
order v exists v if and only if v ≡ 0 or 1
(mod 3).
S. H. Y. Hung and N. S. Mendelsohn, Directed Triple
Systems, Journal of Combinatorial Theory, Series A, 14
(1973), 310-318.
4. Ordered (Oriented)
Triple Systems
Definition. Lindner and Street (1984) define an ordered
triple as either a 3-circuit or a transitive triple. They then
define an ordered triple system of order v, OTS(v), as a
decomposition of Dv into copies of ordered triples. No
restriction is put on the number of 3-circuits nor on the
number of transitive triples.
C. C. Lindner and A. P. Street, Ordered triple systems and
transitive quasigroups, Ars Combinatoria, 17A (1984),
297-306.
Definition. Micale and Pennisi (1993) also dealt with
ordered triple systems, but independently came up with
the idea. They referred to them as oriented triple systems.
Their study addressed two automorphism questions.
B. Micale and M. Pennisi, Cyclic and Rotational Oriented
Triple Systems, Ars Combinatoria, 35 (1993), 65-68.
Theorem. An ordered (oriented) triple
system of order v exists if and only if
v ≡ 0 or 1 (mod 3).
5. Hybrid Triple
Systems
Definition. Colbourn, Pulleyblank, and Rosa (1989)
define a hybrid triple system of order v, HTS(v), as a
decomposition of Kv into a given number of copies of
3-circuits and transitive triples. That is, a c-HTS(v) is
a decomposition of Dv into c copies of a 3-circuit and
v(v – 1)/3 – c copies of a transitive triple.
C. J. Colbourn, W. R. Pulleyblank, and A. Rosa, Hybrid
Triple Systems and Cubic Feedback Sets, Graphs and
Combinatorics, 5 (1989), 15-28.
Note. When c = 0, a c-HTS(v) is a directed triple
system of order v. When c = v(v – 1)/3, a c-HTS(v) is
a Mendelsohn triple system of order v.
Theorem. A c-HTS(v) exists if and only if v ≡ 0 or 1 (mod
3), v ≠ 6 when c is 9 or 10, and
c

{0, 1, 2, …, v(v – 1)/3 – 2, v(v – 1)/3}.
c
v(v – 1)/3 – c
Note. Heinrich (1991) gave direct constructions for cHTS(v) and solved the problem for λ-fold systems.
K. Heinrich, Simple Direct Constructions for Hybrid Triple
Designs, Discrete Mathematics, 97 (1991), 223-227.
6. The Last of the
Triple Systems
Note. Hartman and Mendelsohn (1986)
considered decompositions of the complete
directed graph, Dv, into every possible digraph on
3 vertices. In fact, they solved the problem for λfold complete digraphs.
A. Hartman and E. Mendelsohn, The Last of the Triple
Systems, Ars Combinatoria, 22 (1986), 25-41.
Note. There are 13 different simple digraphs on 3 vertices:
G3
D3
G5
P2
T4
M4
M3
T3
T c4
Gc3
M2
T2
c
T2
7. Mixed Triple
Systems
Definition. A mixed graph consists of a vertex set, and
edge set, and an arc set. The complete mixed graph on
v vertices, Mv, has an edge between every two vertices
and an arc from every vertex to every other vertex.
M4
Note. There are 3 partial orientations of the 3-cycle
with two arcs and one edge:
T1
T2
T3
Definition. A decomposition of Mv into copies of Ti is a
Ti-mixed triple system of order v.
Theorem. A Ti-triple system of order v exists if and
only if v ≡ 1 (mod 2) for i =1, 2, 3, except for v = 3, 5
when i = 3.
T1
T2
T3
R. Gardner, Triple Systems from Mixed Graphs, Bulletin
of the ICA, 27 (1999), 95-100.
Note. Inspired by Hartman and Mendelsohn (“The
Last of the Triple Systems”), we are lead to
consider all possible mixed graphs on 3 vertices.
There are 18 such mixed graphs with (like Mv)
twice as many arcs as edges.
T
T
T
2
1
T
2
2
T
2
3
T
2
5
T
2
6
T
2
7
T
2
8
T
4
3
2
4
4
2
T
T
T
4
6
T
4
7
T
T
4
4
T
6
1
4
1
4
5
4
8
T
4
9
Note. Decompositions of Mv into each of the 18 mixed
graphs above is currently being studied by Ernest Jum as
part of his master’s thesis at East Tennessee State
University.
Current Progress. Mr. Jum has one case left (T46 and its
converse) when λ = 1. The thesis will be titled “The
Last of the Mixed Triple Systems.”
Some Observations about T44 Decompositions
4
There is a T4 -decomposition of M4. For v ≡ 1 or 4 (mod
12), there is an M4 decomposition of Dv (Hanani). So there
is a T44 -decomposition of Dv for all v ≡ 1 or 4 (mod 12).
T44
H. Hanani, Balanced incomplete block designs and related
designs, Discrete Mathematics, 11 (1975), 255-369.
Some Observations about T44 Decompositions (cont.)
One can see from the decomposition of M4 into copies of
T44 , that the “mixed wheel” can be decomposed into copies
of T44 also. Hartman and Mendelsohn used wheels
extensively in some of their constructions, and similar
constructions are used in additional T44 – decompositions.
T44
8. Hybrid Triple
Systems from
Digraph-Pair
Multidecompositions
Definition. A graph-pair of order t is two nonisomorphic graphs G and H on t (non-isolated) vertices
for which G U H = Kt . A decomposition of Kv into a
collection of copies of G and copies of H, where at
least one copy of each is present, is a (G,H)multidecomposition of Kv (Abueida and Devan).
G
H
A. Abueida and M. Devan, Multidesigns for Graph-Pairs of Order 4
and 5, Graphs and Combinatorics, 19 (2003), 433-447.
Note. We define a digraph-pair similar to the definition of
a graph-pair and concentrate on digraph-pairs of order 3.
This gives three such pairs.
G1
H1
G2
G3
H2
H3
Note. We now introduce the concept of a hybrid triple
system based on digraph multidecompositions. We study
(Gi , Hi)-multidecompositions of Dv which consist of gi
copies of Gi and hi = (v(v – 1)/3 – gi)/2 copies of Hi , for
all possible values of gi and hi (for i {1, 2, 3}). We call
such a decomposition a gi-hybrid triple system of type i
and order v.
This work is being done jointly by Beeler and Gardner.
Lemma. Each vertex of G1 and each vertex of H1 has outdegree even, so a necessary condition for the existence of
a g1-hybrid triple system of order v is that v ≡ 1 (mod 2).
G1
H1
Note. G1 can be decomposed into two copies of H1 .
G1
H1 U H1
Theorem. A G1-decomposition of Dv exists iff v ≡ 1 (mod
4) (Hartman and Mendelsohn).
Theorem. When v ≡ 1 (mod 4), a g1-hybrid triple
system of order v exists iff g1 
{0, 1, 2, …, v(v – 1)/4}
and h1 = (v(v – 1) – 4g1)/2.
Lemma. A necessary condition for a g1-hybrid triple system
of order v ≡ 3 (mod 4) is h1 ≥ v/3 .
Proof. Suppose not. Then there is some vertex x in no
copy of H1. For x to have out-degree v – 1 (even), x must
be in exactly (v – 1)/2 blocks of the following form:
x
In the union of these blocks, x has total indegree (v – 1)/2 ≡ 1 (mod 2). But in the
remaining blocks x is of in-degree even.
So in the collection of G1s, x is of total in-degree odd,
contradiction. Therefore every vertex of Dv must be in at
least one copy of H1.
Conjecture. A g1-hybrid triple system of order v exists iff

• v ≡ 1 (mod 4) and g1 {0, 1, 2, …, v(v – 1)/4} and
h1 = (v(v – 1) – 4g1)/2, or
• v ≡ 3 (mod 4) and h1 { v/3 , v/3 +1, …, v(v – 1)/4}
and g1 = (v(v – 1) – 2h1)/4.
Note. Since G1 and G2 are converses, the existence of a
g2-hybrid triple system will follow from the existence of
a g1-hybrid triple system.
Current Progress. With the possible exception of
some small cases, the existence of g3-hybrid triple
systems is mostly settled.