\(D\)-optimal exact designs in a completely randomized statistical set-up are constructed, for comparing \(n > 2\) qualitative factors (treatments), making \(r\) observations per treatment level in the presence of \(n\) (or less) quantitative or continuous factors (regression factors or covariates) of influence. Their relation with cyclic supplementary difference sets \(2-{(u; k_1, k_2; \lambda)}\) is shown, when \(n = 2u \equiv 2 \pmod{4}\), \(r \equiv 1 \pmod{2}\), \(r \neq 1\), \(r < u\) and \(k_1, k_2, \lambda\) are defined by \(1 \leq k_1 \leq k_2 \leq (u-1)/2\), \((u-2k_1)^2 + (u-2k_2)^2 = 2(ur+u-r)\), \(\lambda = k_1 + k_2 – (u-r)/2\). Making use of known cyclic difference sets, the existence of a multiplier and the non-periodic autocorrelation function of two sequences, such supplementary difference sets are constructed for the first time. A list of all 201 supplementary difference sets \(2-{(u; k_1, k_2; \lambda)}\) for \(n = 2u < 100\) is given.

In this paper, we consider a permutation \(\sigma \in S_n\) as acting on an arbitrary tree with \(n\) vertices (labeled \(1, 2, 3, \ldots, n\)). Each edge \([a, b]\) of \(T\) corresponds to a transposition \((a, b) \in S_n\), and such a “tree of transpositions” forms a minimal generating set for \(S_n\). If \(\sigma \in S_n\), then \(\sigma\) may be written as a product of transpositions from \(T, \sigma = t_k t_{k-1} \ldots t_2t_1\). We will refer to such a product as a \(T\)-factorization of \(\sigma\) of length \(k\). The primary purpose of this paper is to describe an algorithm for producing \(T\)-factorizations of \(\sigma\). Although the algorithm does not guarantee minimal factorizations, both empirical and theoretical results indicate that the factorizations produced are “nearly minimal”. In particular, the algorithm produces factorizations that never exceed the known upper bounds.

The linear vertex-arboricity of a surface \(S\) is the maximum of the linear vertex-arboricities of all graphs embeddable into \(S\). Poh showed that the linear vertex-arboricity of a sphere is three. We show that the linear vertex-arboricities of a projective plane and a torus are three and four, respectively. Moreover, we show that the linear vertex-arboricity of a Klein bottle is three or four.