Types of directed triple systems

We introduce three types of directed triple systems. Two of these, Mendelsohn directed triple systems and Latin directed triple systems, have previously appeared in the literature but we prove further results about them. The third type, which we call skewed directed triple systems, is new and we determine the existence spectrum to be v ≡ 1 (mod 3), v ≠ 7, except possibly for v = 22, as well as giving enumeration results for small orders

Latin directed triple systems

Abstract It is well known that given a Steiner triple system then a quasigroup can be formed by defining an operation · by the identities x · x = x and x · y = z where z is the third point in the block containing the pair {x, y}. The same is true for a Mendelsohn triple system where the pair (x, y) is considered to be ordered. But it is not true in general for directed triple systems. However directed triple systems which form quasigroups under this operation do exist. We call these Latin directed triple systems and in this paper begin the study of their existence and properties

The intersection spectrum of Skolem sequences and its applications to lambda fold cyclic triple systems, together with the Supplement

A Skolem sequence of order n is a sequence S_n=(s_{1},s_{2},...,s_{2n}) of 2n integers containing each of the integers 1,2,...,n exactly twice, such that two occurrences of the integer j in {1,2,...,n} are separated by exactly j-1 integers. We prove that the necessary conditions are sufficient for existence of two Skolem sequences of order n with 0,1,2,...,n-3 and n pairs in same positions. Further, we apply this result to the fine structure of cyclic two, three and four-fold triple systems, and also to the fine structure of lambda-fold directed triple systems and lambda-fold Mendelsohn triple systems. For a better understanding of the paper we added more details into a "Supplement".Comment: The Supplement for the paper "The intersection spectrum of Skolem sequences and its applications to lambda fold cyclic triple systems" is available here. It comes right after the paper itsel

Block-avoiding point sequencings

Let $n$ and $\ell$ be positive integers. Recent papers by Kreher, Stinson and Veitch have explored variants of the problem of ordering the points in a triple system (such as a Steiner triple system, directed triple system or Mendelsohn triple system) on $n$ points so that no block occurs in a segment of $\ell$ consecutive entries (thus the ordering is locally block-avoiding). We describe a greedy algorithm which shows that such an ordering exists, provided that $n$ is sufficiently large when compared to $\ell$. This algorithm leads to improved bounds on the number of points in cases where this was known, but also extends the results to a significantly more general setting (which includes, for example, orderings that avoid the blocks of a design). Similar results for a cyclic variant of this situation are also established. We construct Steiner triple systems and quadruple systems where $\ell$ can be large, showing that a bound of Stinson and Veitch is reasonable. Moreover, we generalise the Stinson--Veitch bound to a wider class of block designs and to the cyclic case. The results of Kreher, Stinson and Veitch were originally inspired by results of Alspach, Kreher and Pastine, who (motivated by zero-sum avoiding sequences in abelian groups) were interested in orderings of points in a partial Steiner triple system where no segment is a union of disjoint blocks. Alspach~\emph{et al.}\ show that, when the system contains at most $k$ pairwise disjoint blocks, an ordering exists when the number of points is more than $15k-5$. By making use of a greedy approach, the paper improves this bound to $9k+O(k^{2/3})$.Comment: 38 pages. Typo in the statement of Theorem 17 corrected, and other minor changes mad

Resolvable Mendelsohn designs and finite Frobenius groups

We prove the existence and give constructions of a $(p(k)-1)$-fold perfect resolvable $(v, k, 1)$-Mendelsohn design for any integers $v > k \ge 2$ with $v \equiv 1 \mod k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $\phi$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K \rtimes \langle \phi \rangle$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v = p_{1}^{e_1} \ldots p_{t}^{e_t} \ge 3$ in prime factorization, and any prime $k$ dividing $p_{i}^{e_i} - 1$ for $1 \le i \le t$, there exists a resolvable perfect $(v, k, 1)$-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if $k$ is even and divides $p_{i} - 1$ for $1 \le i \le t$, then there are at least $\varphi(k)^t$ resolvable $(v, k, 1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $\varphi$ is Euler's totient function.Comment: Final versio

Decompositions of the Complete Mixed Graph by Mixed Stars

In the study of mixed graphs, a common question is: What are the necessary and suffcient conditions for the existence of a decomposition of the complete mixed graph into isomorphic copies of a given mixed graph? Since the complete mixed graph has twice as many arcs as edges, then an obvious necessary condition is that the isomorphic copies have twice as many arcs as edges. We will prove necessary and suffcient conditions for the existence of a decomposition of the complete mixed graphs into mixed stars with two edges and four arcs. We also consider some special cases of decompositions of the complete mixed graph into partially oriented stars with twice as many arcs as edges. We employ difference methods in most of our constructions when showing suffciency.

Decompositions of Mixed Graphs with Partial Orientations of the P\u3csub\u3e4\u3c/sub\u3e.

A decomposition D of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. A mixed graph on V vertices is an ordered pair (V,C), where V is a set of vertices, |V| = v, and C is a set of ordered and unordered pairs, denoted (x, y) and [x, y] respectively, of elements of V [8]. An ordered pair (x, y) ∈ C is called an arc of (V,C) and an unordered pair [x, y] ∈ C is called an edge of graph (V,C). A path on n vertices is denoted as Pn. A partial orientation on G is obtained by replacing each edge [x, y] ∈ E(G) with either (x, y), (y, x), or [x, y] in such a way that there are twice as many arcs as edges. The complete mixed graph on v vertices, denoted Mv, is the mixed graph (V,C) where for every pair of distinct vertices v1, v2 ∈ V , we have {(v1, v2), (v2, v1), [v1, v2]} ⊂ C. The goal of this thesis is to establish necessary and sufficient conditions for decomposition of Mv by all possible partial orientations of P4

Decompositions of Complete Symmetric Directed Graphs into the Oriented Heptagons

The complete symmetric directed graph of order $v$, denoted $K_{v}^*$, is the directed graph on $v$ vertices that contains both arcs $(x,y)$ and $(y,x)$ for each pair of distinct vertices $x$ and $y$. For a given directed graph, $D$, the set of all $v$ for which $K_{v}^*$ admits a $D$-decomposition is called the spectrum of $D$. There are 10 non-isomorphic orientations of a $7$-cycle (heptagon). In this paper, we completely settled the spectrum problem for each of the oriented heptagons.Comment: 10 pages, 1 figur