Twofold Triple Systems Research Articles

Decomposable Twofold Triple Systems with Non-Hamiltonian 2-Block Intersection Graphs

The $2$-block intersection graph ($2$-BIG) of a twofold triple system (TTS) is the graph whose vertex set is composed of the blocks of the TTS and two vertices are joined by an edge if the corresponding blocks intersect in exactly two elements. The $2$-BIGs are themselves interesting graphs: each component is cubic and $3$-connected, and a $2$-BIG is bipartite exactly when the TTS is decomposable to two Steiner triple systems. Any connected bipartite $2$-BIG with no Hamilton cycle is a further counter-example to a disproved conjecture posed by Tutte in 1971. Our main result is that there exists an integer $N$ such that for all $v\geq N$, if $v\equiv 1$ or $3\mod{6}$ then there exists a TTS($v$) whose $2$-BIG is bipartite and connected but not Hamiltonian. Furthermore, $13<N\leq 663$. Our approach is to construct a TTS($u$) whose $2$-BIG is connected bipartite and non-Hamiltonian and embed it within a TTS($v$) where $v>2u$ in such a way that, after a single trade, the $2$-BIG of the resulting TTS($v$) is bipartite connected and non-Hamiltonian.

Twofold triple systems with cyclic 2‐intersecting Gray codes

AbstractGiven a combinatorial design with block set , the block‐intersection graph (BIG) of is the graph that has as its vertex set, where two vertices and are adjacent if and only if . The i‐block‐intersection graph (i‐BIG) of is the graph that has as its vertex set, where two vertices and are adjacent if and only if . In this paper, several constructions are obtained that start with twofold triple systems (TTSs) with Hamiltonian 2‐BIGs and result in larger TTSs that also have Hamiltonian 2‐BIGs. These constructions collectively enable us to determine the complete spectrum of TTSs with Hamiltonian 2‐BIGs (equivalently TTSs with cyclic 2‐intersecting Gray codes) as well as the complete spectrum for TTSs with 2‐BIGs that have Hamilton paths (i.e. for TTSs with 2‐intersecting Gray codes). In order to prove these spectrum results, we sometimes require ingredient TTSs that have large partial parallel classes; we prove lower bounds on the sizes of partial parallel classes in arbitrary TTSs, and then construct larger TTSs with both cyclic 2‐intersecting Gray codes and parallel classes.

Twofold triple systems without 2-intersecting Gray codes

Given a combinatorial design $$\mathcal {D}$$D with block set $$\mathcal {B}$$B, the block-intersection graph (BIG) of $$\mathcal {D}$$D is the graph having $$\mathcal {B}$$B as its vertex set, and in which two vertices $$B_{1} \in \mathcal {B}$$B1źB and $$B_{2} \in \mathcal {B} $$B2źB are adjacent if and only if $$|B_{1} \cap B_{2}| > 0$$|B1źB2|>0. The i-block-intersection graph (i-BIG) of $$\mathcal {D}$$D is the graph having $$\mathcal {B}$$B as its vertex set, and in which two vertices $$B_{1} \in \mathcal {B}$$B1źB and $$B_{2} \in \mathcal {B}$$B2źB are adjacent if and only if $$|B_{1} \cap B_{2}| = i$$|B1źB2|=i. In this paper we present several constructions which together enable us to determine the complete spectrum of twofold triple systems with connected non-Hamiltonian 2-BIGs (equivalently, the complete spectrum of twofold triple systems that have no cyclic 2-intersecting Gray codes but for which the 2-BIGs are nevertheless connected); this spectrum consists of all orders $$v \equiv 0$$vź0 or 1 (modulo 3) such that $$v \geqslant 6$$vź6, except for $$v \in \{7,9,10\}$$vź{7,9,10}. We also determine all but a finite number of the elements of the spectrum for twofold triple systems for which the 2-BIGs are connected but have no Hamilton path (i.e., for systems which lack 2-intersecting Gray codes but nevertheless have connected 2-BIGs); specifically, the spectrum is found to consist of every order $$v \equiv 0$$vź0 or 1 (modulo 3) such that $$v \geqslant 13$$vź13, except possibly for $$v \in \{13,15,16,27,28,30,31,33,34,37 \}$$vź{13,15,16,27,28,30,31,33,34,37}.

Further biembeddings of twofold triple systems

We construct face two-colourable triangulations of the graph 2 K n in an orientable surface; equivalently biembeddings of two twofold triple systems of order n , for all n ≡ 16 or 28 (mod 48) . The biembeddings come from index 1 current graphs lifted under a group Z n / 4 × K 4 .

Cyclic Biembeddings of Twofold Triple Systems

We construct face two-colourable triangulations of the graph 2Kn in an orientable surface; equivalently biembeddings of two twofold triple systems of order n, for all n ≡ 4 (mod 12). The biembeddings have a cyclic automorphism and the construction employs index 1 current graphs.

Metamorphosis of simple twofold triple systems into maximum twofold [formula omitted]-packings

Let (X,B) be a simple twofold triple system of order v. For every x,y∈X, x≠y, the pair {x,y} is contained in exactly two different triples, say, {x,y,z} and {x,y,w}. Any two blocks B1,B2∈B satisfying |B1∩B2|=2 form a matched pair. Suppose that there is a partition of B into |B|/2 matched pairs. If we replace the double edge {x,y} with its corresponding single edge {x,y} from a matched pair {x,y,z}, {x,y,w}, we have a (K4−e)[x,y,z−w]. Let C be the collection of (K4−e)s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and F be the collection of the deleted edges. If F can be reassembled into a collection D of ⌊v(v−1)/30⌋(K4−e)s, then (X,C∪D) is a maximum twofold (K4−e)-packing of order v. We call (X,C∪D) a metamorphosis of the simple twofold triple system (X,B). In this paper, we show that there exists a metamorphosis of a simple twofold triple system of order v into a maximum twofold (K4−e)-packing of order v if and only if v≡0,1(mod3) and v≥4 with two exceptions of v=6,7 and one possible exception of v=18.

Metamorphosis of simple twofold triple systems into twofold -designs

Abstract In this note, we show that there exists a metamorphosis of a simple twofold triple system of order v into a twofold ( K 4 − e ) -design of order v if and only if v ≡ 0 , 1 , 6 , 10 ( mod 15 ) and v ⩾ 10 .

Triple metamorphosis of twofold triple systems

In a simple twofold triple system (X,B), any two distinct triples T1, T2 with |T1∩T2|=2 form a matched pair. Let F be a pairing of the triples of B into matched pairs (if possible). Let D be the collection of double edges belonging to the matched pairs in F, and let F∗ be the collection of 4-cycles obtained by removing the double edges from the matched pairs in F. If the edges belonging to D can be assembled into a collection of 4-cycles D∗, then (X,F∗∪D∗) is a twofold 4-cycle system called a metamorphosis of the twofold triple system (X,B). Previous work (Gionfriddo and Lindner, 2003 [7]) has shown that the spectrum for twofold triple systems having a metamorphosis into a twofold 4-cycle system is precisely the set of all n≡0,1,4 or 9(mod12), n≥9. In this paper, we extend this result as follows. We construct for each n≡0,1,4 or 9(mod12), n≠9 or 12, a twofold triple system (X,B) with the property that the triples in B can be arranged into three sets of matched pairs F1, F2, F3 having metamorphoses into twofold 4-cycle systems (X,F1∗∪D1∗), (X,F2∗∪D2∗), and (X,F3∗∪D3∗), respectively, with the property that D1∪D2∪D3=2Kn. In this case we say that (X,B) has a triple metamorphosis. Such a twofold triple system does not exist for n=9, and its existence for n=12 remains an open and apparently a very difficult problem.

Triangular embeddings of complete graphs (neighborly maps) with 12 and 13 vertices

AbstractIn this paper, we describe the generation of all nonorientable triangular embeddings of the complete graphs K12 and K13. (The 59 nonisomorphic orientable triangular embeddings of K12 were found in 1996 by Altshuler, Bokowski, and Schuchert, and K13 has no orientable triangular embeddings.) There are 182,200 nonisomorphic nonorientable triangular embeddings for K12, and 243,088,286 for K13. Triangular embeddings of complete graphs are also known as neighborly maps and are a type of twofold triple system. We also use methods of Wilson to provide an upper bound on the number of simple twofold triple systems of order n, and thereby on the number of triangular embeddings of Kn. We mention an application of our results to flexibility of embedded graphs. © 2005 Wiley Periodicals, Inc. J Combin Designs

Simple and indecomposable twofold cyclic triple systems from Skolem sequences

Simple and indecomposable twofold cyclic triple systems from Skolem sequences

Case studies and new results in combinatorial enumeration

A 1976 block design enumeration algorithm has been reimplemented with a number of enhancements. Its use is demonstrated with a number of case studies involving different kinds of incidence structures. New results include the enumeration of weakly union-free twofold triple systems of orders 12 and 13, Mendelsohn triple systems of orders 10 and 12, 2-(8, 4, 9), 2-(10, 4, 4), 2-(16, 4, 2), 3-(11, 5, 4) and 4-(12, 6, 4) block designs, and 2-(7, 3, λ) designs for 14 ≤ λ ≤ 16. Some of these results correct enumerations previously published in the literature. Additional results confirm independently produced results, some of which have been published quite recently. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 239–260, 2000

Case studies and new results in combinatorial enumeration

A 1976 block design enumeration algorithm has been reimplemented with a number of enhancements. Its use is demonstrated with a number of case studies involving different kinds of incidence structures. New results include the enumeration of weakly union-free twofold triple systems of orders 12 and 13, Mendelsohn triple systems of orders 10 and 12, 2-(8, 4, 9), 2-(10, 4, 4), 2-(16, 4, 2), 3-(11, 5, 4) and 4-(12, 6, 4) block designs, and 2-(7, 3, λ) designs for 14 ≤ λ ≤ 16. Some of these results correct enumerations previously published in the literature. Additional results confirm independently produced results, some of which have been published quite recently. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 239–260, 2000

Weakly union-free twofold triple systems

In this paper, we settle a problem of Frankl and Filredi, which is a special case of a problem of Erdt~s, determining the maximum number of hyperedges in a 3-uniform hypergraph in which no two pairs of distinct hyperedges have the same union. The extremal case corresponds to the existence of weakly union-free twofold triple systems, which is settled here with six definite and four possible exceptions. An application to group testing is also given.

Partitions of sets of two-fold triple systems, and their relation to some strongly regular graphs

LetX be a set ofv + 1 elements, wherev ≡ 0 or 1 (mod 3). If two copies of the collection of\(\left( {\begin{array}{*{20}c} {v + 1} \\ 3 \\ \end{array} } \right)\) triples chosen fromX can be partitioned intov + 1 mutually disjoint two-fold triple systems, each based on a differentv-subset ofX, we say they form an overlarge set of two-fold triple systems, denoted byOS(TTS(v)). In this paper, we give the first construction of anOS(TTS(10)). We then investigate the properties of the uniqueOS(TTS(6)) and obtain: (i) A partition of the set of 84 distinctTTS(6) based onX = {1, 2,..., 7} into 12 parallel classes, that is, into 12OS(TTS(6)) each containing sevenTTS(6); (ii) A resolution of the set of 1008 distinctOS(TTS(6)) based onX into 84 parallel classes; (iii) Simple constructions of several strongly-regular graphs, including both the Hoffman-Singleton and Higman-Sims graphs, which arise from the relation between the family of 84 distinctTTS(6) and the family of 30 distinctSTS(7), all based onX.

Multi-Set designs and numbers of common triples

A two-fold multi-set triple system is a collection of triples chosen, possibly with repetitions, from av-set in such a way that each pair (whether distinct or not) occurs precisely two times. For instance the triplex, x, y (which we shall write asxxy) contains the pairxx once and the pairxy twice. Such a design has v(v + l)/3 blocks, and a necessary and sufficient condition for existence is thatv = 0 or 2 (mod 3) and y≥5. The design is called simple if it contains no repeated blocks. Let I2(v) denote the set of non-negative integerss such that there exist two simple two-fold multi-set triple systems of orderv with preciselys blocks in common. In this paper we determine I2(v) for all admissiblev.

Partitioning twofold triple systems into complete arcs

We establish that for all s, there exists a design with parameters ( s 2,3,2) such that the points can be partitiones into s complete s-arcs. Furthermore we present a general technique which applies to the construction of designs ( s 2,4,1) possessing a partition into s complete s-arcs. This gives a partial solution to this question as well.

Enumeration of small nonisomorphic 1-rotational twofold triple systems

In this paper, twofold triple systems of order v are enumerated for all υ ≤ 19 \upsilon \leq 19 .

On twofold triple systems and their automorphism groups

A general construction of 2 − ( v, 3, 2) designs (twofold triple systems) is used to produce simple indecomposable designs for all admissible v ≠ 3, 7. Such systems with cyclic automorphism groups are directly constructed. Also indecomposable designs with dihedral automorphism groups are constructed.

Element neighbourhoods in twofold triple systems

The method of differences is used to establish that every 2-regular multigraph onv− 1≡0,2 (mod 3) points occurs as the neighbourhood graph of an element in a twofold triple system of orderv, with two exceptions: C2∪C3and C3∪C3.

Repeated blocks in indecomposable twofold triple systems

Repeated blocks in indecomposable twofold triple systems