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 { G 1, G 2, …, G n } where G i G and V(G i ) V(H) for all i, E(G i ) ∩ E(G j ) = Ø if i ≠ j, and G i = H. Note. Decompositions of digraphs and mixed graphs are siimilarly defined.

Example. There is a decomposition of K 5 into 5-cycles.

(0,1,3) (1,2,4) (2,3,5) (3,4,6) (4,5,0) (5,6,1) (6,0,2) Example. There is a decomposition of K 7 into 3-cycles:

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, K v, into 3-cycles.

From the Saint Andrews MacTutor History of Mathematics website. Jakob Steiner J. Steiner, Combinatorische Aufgabe, Journal für die Reine und angewandte Mathematik (Crelle’s Journal), 45 (1853), v ≡ 1 or 3 (mod 6) is necessary.

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), Theorem. A STS(v) exists if and only if v ≡ 1 or 3 (mod 6). Note. Sufficiency follows from Reiss.

Thomas P. Kirkman From the Saint Andrews MacTutor History of Mathematics website. T. Kirkman, On a problem in combinations, Cambridge and Dublin Mathematics Journal, 2 (1847), STS(v) iff v ≡ 1 or 3 (mod 6).

R. Peltesohn, Eine Lösung der beiden Heffterschen Differenzenprobleme, Compositio Math., 6 (1939), Constructions Based on Difference Methods L. Heffter, Ueber Triplesysteme, Math. Ann., 49 (1897), Heffter posed two difference problems: The Problems were solved by Peltesohn:

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 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),

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),

4. Ordered (Oriented) Triple Systems

C. C. Lindner and A. P. Street, Ordered triple systems and transitive quasigroups, Ars Combinatoria, 17A (1984), 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 D v into copies of ordered triples. No restriction is put on the number of 3-circuits nor on the number of transitive triples.

B. Micale and M. Pennisi, Cyclic and Rotational Oriented Triple Systems, Ars Combinatoria, 35 (1993), 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.

Theorem. An ordered (oriented) triple system of order v exists if and only if v ≡ 0 or 1 (mod 3).

5. Hybrid Triple Systems

C. J. Colbourn, W. R. Pulleyblank, and A. Rosa, Hybrid Triple Systems and Cubic Feedback Sets, Graphs and Combinatorics, 5 (1989), Definition. Colbourn, Pulleyblank, and Rosa (1989) define a hybrid triple system of order v, HTS(v), as a decomposition of K v into a given number of copies of 3-circuits and transitive triples. That is, a c-HTS(v) is a decomposition of D v into c copies of a 3-circuit and v(v – 1)/3 – c copies of a transitive triple. 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.

K. Heinrich, Simple Direct Constructions for Hybrid Triple Designs, Discrete Mathematics, 97 (1991), 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}. Note. Heinrich (1991) gave direct constructions for c- HTS(v) and solved the problem for λ-fold systems. c v(v – 1)/3 – c

6. The Last of the Triple Systems

A. Hartman and E. Mendelsohn, The Last of the Triple Systems, Ars Combinatoria, 22 (1986), Note. Hartman and Mendelsohn (1986) considered decompositions of the complete directed graph, D v, into every possible digraph on 3 vertices. In fact, they solved the problem for λ- fold complete digraphs.

DG GG T T TT T P MM M c c c Note. There are 13 different simple digraphs on 3 vertices:

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, M v, has an edge between every two vertices and an arc from every vertex to every other vertex. M4M4

Note. There are 3 partial orientations of the 3-cycle with two arcs and one edge: T TT 12 3 Definition. A decomposition of M v into copies of T i is a T i -mixed triple system of order v.

Theorem. A T i -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. T TT 12 3 R. Gardner, Triple Systems from Mixed Graphs, Bulletin of the ICA, 27 (1999),

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 M v ) twice as many arcs as edges.

T TT T T T T T T T TTT TT T T T 4 1

Note. Decompositions of M v 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 (T 4 and its converse) when λ = 1. The thesis will be titled “The Last of the Mixed Triple Systems.” 6

Some Observations about T 4 Decompositions There is a T 4 -decomposition of M 4. For v ≡ 1 or 4 (mod 12), there is an M 4 decomposition of D v (Hanani). So there is a T 4 -decomposition of D v for all v ≡ 1 or 4 (mod 12). H. Hanani, Balanced incomplete block designs and related designs, Discrete Mathematics, 11 (1975), T4T

Some Observations about T 4 Decompositions (cont.) One can see from the decomposition of M 4 into copies of T 4, that the “mixed wheel” can be decomposed into copies of T 4 also. Hartman and Mendelsohn used wheels extensively in some of their constructions, and similar constructions are used in additional T 4 – decompositions. T4T

8. Hybrid Triple Systems from Digraph-Pair Multidecompositions

A. Abueida and M. Devan, Multidesigns for Graph-Pairs of Order 4 and 5, Graphs and Combinatorics, 19 (2003), Definition. A graph-pair of order t is two non- isomorphic graphs G and H on t (non-isolated) vertices for which G U H = K t. A decomposition of K v 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 K v (Abueida and Devan). GH

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. G1G1 G2G2 G3G3 H1H1 H2H2 H3H3

Note. We now introduce the concept of a hybrid triple system based on digraph multidecompositions. We study (G i, H i )-multidecompositions of D v which consist of g i copies of G i and h i = (v(v – 1)/3 – g i )/2 copies of H i, for all possible values of g i and h i (for i {1, 2, 3}). We call such a decomposition a g i -hybrid triple system of type i and order v. This work is being done jointly by Beeler and Gardner.

Lemma. Each vertex of G 1 and each vertex of H 1 has out- degree even, so a necessary condition for the existence of a g 1 -hybrid triple system of order v is that v ≡ 1 (mod 2). G1G1 H1H1

Note. G 1 can be decomposed into two copies of H 1. G1G1 H 1 U H 1 Theorem. A G 1 -decomposition of D v exists iff v ≡ 1 (mod 4) (Hartman and Mendelsohn). Theorem. When v ≡ 1 (mod 4), a g 1 -hybrid triple system of order v exists iff g 1 {0, 1, 2, …, v(v – 1)/4} and h 1 = (v(v – 1) – 4g 1 )/2.

Lemma. A necessary condition for a g 1 -hybrid triple system of order v ≡ 3 (mod 4) is h 1 ≥ v/3. Proof. Suppose not. Then there is some vertex x in no copy of H 1. 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 in- degree (v – 1)/2 ≡ 1 (mod 2). But in the remaining blocks x is of in-degree even. So in the collection of G 1 s, x is of total in-degree odd, contradiction. Therefore every vertex of D v must be in at least one copy of H 1.

Conjecture. A g 1 -hybrid triple system of order v exists iff v ≡ 1 (mod 4) and g 1 {0, 1, 2, …, v(v – 1)/4} and h 1 = (v(v – 1) – 4g 1 )/2, or v ≡ 3 (mod 4) and h 1 { v/3, v/3 +1, …, v(v – 1)/4} and g 1 = (v(v – 1) – 2h 1 )/4.

Note. Since G 1 and G 2 are converses, the existence of a g 2 -hybrid triple system will follow from the existence of a g 1 -hybrid triple system. Current Progress. With the possible exception of some small cases, the existence of g 3 -hybrid triple systems is mostly settled.