A balanced part ternary design (BPTD) is a balanced ternary design (BTD) with a specified number of blocks, say \(b_2\), each having repeated elements. We prove some necessary conditions on \(b_2\) and show the existence of some particular BPTDs. We also give constructions of infinite families of BPTDs with \(b_1 = 0\), including families of ternary \(t\)-designs with \(t \geq 3\).

We prove a very natural generalization of the Borsuk-Ulam antipodal theorem and deduce from it, in a very straightforward way, the celebrated result of Alon [1] on splitting necklaces. Alon’s result states that \(t(k-1)\) is an upper bound on the number of cutpoints of an opened \(t\)-colored necklace such that the segments obtained can be used to partition the set of vertices of the necklace into $k$ subsets with the property that every color is represented by the same number of vertices in any element of the partition. The proof of our generalization of the Borsuk-Ulam theorem uses a result from algebraic topology as a starting point and is otherwise purely combinatorial.

Two classical theorems about tournaments state that a tournament with no less than eight vertices admits an antidirected Hamiltonian path and an even cardinality tournament with no less than sixteen vertices admits an antidirected Hamiltonian cycle. Sequential algorithms for finding such a path as well as a cycle follow directly from the proofs of the theorems. Unfortunately, these proofs are inherently sequential and cannot be exploited in a parallel context. In this paper, we propose new proofs leading to efficient parallel algorithms.

In this article, we discuss the number of pairwise orthogonal Latin squares and obtain the estimate \(n_r < 8(r + 1)2^{4r}\) for \(r \geq 2\).

In this paper, we present some results on the existence of balanced arrays (B-arrays) with two symbols and of strength four by using some inequalities involving the statistical concepts of skewness and kurtosis. We demonstrate also, through an illustrative example, that in certain situations, the results given here lead to sharper upper bounds on the number of constraints for B-arrays.

If \(\alpha\) is a primitive root of the finite field \({GF}(2^n)\), we define a function \(\pi_n\) on the set \({E}_n = \{1, 2, \ldots, 2^n – 2\}\) by
\[
\pi_\alpha(i) = j \quad \text{iff} \quad \alpha^i = 1 + \alpha^{j}.
\]
Then \(\pi_\alpha\) is a permutation of \({E}_n\) of order \(2\). The path-length of \(\pi\), denoted \({PL}(\pi)\), is the sum of all the quantities \(|\pi(i) – i|\),
and the rank of \(\pi\) is the number of pairs \((i, j)\) with \(i \pi(j)\). We show that \({PL}(\pi) = {2(2^n – 1)(2^{n-1} – 1)}/{3}\), and the rank of \(\pi\) is \((2^{n-1} – 1)^2\).

If \(\gcd(k, 2^n – 1) = 1\), then \(M_k(x) = kx(\mod{2^n – 1})\) is a permutation of \({E}_n\). We show that a necessary condition for the function \(f_i(x) = 1 + x + \cdots + x^{i}\) to be a permutation of \({GF}(2^n)\), is that the function \(g_k(r) = \pi(M_{k+1}(r)) – \pi(r)\) be a permutation of \({E}_n\) such that exactly half the members \(r\) of \({E}_n\) satisfy \(g_k(r) r\).

Let \((G, \cdot)\) be a group with identity element \(e\) and with a unique element \(h\) of order \(2\). In connection with an investigation into the admissibility of linear groups, one of the present authors was recently asked if, for every cyclic group \(G\) of even order greater than \(6\), there exists a bijection \(\gamma\)
from \(G \setminus \{e, h\}\) to itself such that the mapping \(\delta: g \to g \cdot \gamma(g)\) is again a bijection from \(G \setminus \{e, h\}\) to itself. In the present paper, we answer the above question in the affirmative and we prove the
more general result that every abelian group which has a cyclic Sylow \(2\)-subgroup of order greater than \(6\) has such a partial bijection.

A directed triple system of order \(v\) and index \(\lambda\),denoted \({DTS}_\lambda(v)\), is said to be reverse if it admits an automorphism consisting of \(v/2\) transpositions when \(v\) is even, or a fixed point and \((v-1)/2\) transpositions when \(v\) is odd. We give necessary and sufficient conditions for the existence of a reverse \({DTS}_\lambda(v)\) for all \(\lambda \geq 1\).

A \(1\)-\emph{factor} of a graph \(G\) is a \(1\)-regular spanning subgraph of \(G\).A graph \(G\) has exactly \(t\) \(1\)-factors if the maximum set of edge-disjoint \(1\)-factors is \(t\). For given non-negative integers \(d\), \(t\), and even \(e\), let \(\mathcal{G}(2n; d, e, t)\) be the class of simple connected graphs on \(2n\) vertices, \((2n-1)\) of which have degree \(d\) and one has degree \(d+e\),having exactly \(t\) \(1\)-factors. The problem that arises is that of determining when \(\mathcal{G}(2n; d, e, t) \neq \emptyset ?\) Recently, we resolved the case \(t = 0\). In this paper, we will consider the case \(t = 1\).

In this paper we show that the complete graph \(K_{12}\) is not decomposable into three factors of diameter two, thus resolving a longstanding open problem. This result completes the solution of decomposition of a complete graph into three
factors, one of which has diameter two and the other factors have finite diameters.