We consider a linear model for the comparison \(V \geq 2\) treatments (or one treatment at \(V\) levels) in a completely randomized statistical setup, making \(r\) (the replication number) observations per treatment level in the presence of \(K\) continuous covariates with values on the \(K\)-cube. The main interest is restricted to cyclic designs characterized by the property that the allocation matrix of each treatment level is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. The \(D\)-optimality criterion is used for estimating all the parameters of this model.

By studying the nonperiodic autocorrelation function of circulant matrices, we develop an exhaustive algorithm for constructing \(D\)-optimal cyclic designs with even replication number. We apply this algorithm for \(r = 4, 16 \leq V \leq 24, N=rV \equiv 0 \mod 4\), for \(r=6, 12\leq V \leq 24, N =rV \equiv 0 \mod 4\), for \(r =6, V =m.n, m\) is a prime, \(N =rV \equiv 2 \mod 4\) and the corresponding cyclic designs are given.

New sufficient conditions for equality of edge-connectivity and minimum degree of graphs are presented, including those of Chartrand, Lesniak, Plesnik, Plesník, and Ždmán, and Volkmann.

We show that \((81, 16, 3)\)-block designs have no involutionary automorphisms that fix just \(13\) points. Since the nonexistence of \((81, 16, 3)\)-designs with involutionary automorphism fixing \(17\) points has already been proved, it follows that any involution that an \((81, 16, 3)\)-design may have must fix just \(9\) points.

In this paper we construct a latin \((n \times n \times (n-d))\)-parallelepiped that cannot be extended to a latin cube of order \(n\) for any pair of integers \(d, n\) where \(d \geq 3\) and \(n \geq 2d+1\). For \(d = 2\), it is similar to the construction already known.

In this note the numbers of common triples in two simple balanced ternary designs with block size \(3\), index \(3\) and \(p_2 = 3\) are determined.

The class of parity graphs, those in which the cardinality of every maximal independent subset of vertices has the same parity, contains the well-covered graphs and arose in connection with the PSPACE-complete game “Generalized Kayles”. In 1983 [5] we characterized parity graphs of girth 8 or more. This is extended to a characterization of the parity graphs of girth greater than 5. We deduce that these graphs can be recognized in polynomial time.

In this paper we bring out more strongly the connection between the disconnection number of a graph and its cycle rank. We also show how to associate with a pizza sliced right across in a certain way with \(n-2\) cuts a graph with \(n\) vertices, and show that if the pizza is cut thereby into \(r\) pieces, then any set of \(r-1\) of these pieces corresponds to a basis for the cycle space of the associated graph. Finally we use this to explain why for \(n\geq 3\) the greatest number of regions that can be formed by slicing a pizza in the certain way with \(n-2\) cuts, namely \(\frac{1}{2}(n^2-3n+4)\), equals the disconnection number of \(K_n\).

For a graph \(G\), let \(\sigma_k = \min \{\sum_{i=1}^{k} d(v_i) \mid \{v_1, \ldots, v_k\}\) { is an independent set
of vertices in } G\}. Jung proved that every \(1\)-tough graph \(G\) with \(|V(G)| = n \geq 11\) and \(\sigma_2 > n-4\) is hamiltonian. This result is generalized as follows: if \(G\) is a \(1\)-tough graph with \(|V(G)| = n \geq 3\) such that \(\sigma_3 > n\) and for all \(x, y \in V(G)\), \(d(x,y) = 2\) implies \(\max\{d(x), d(y)\} \geq \frac{1}{2}(n-4)\), then \(G\) is hamiltonian. It is also shown that the condition \(\sigma_3 \geq n\), in the latter result, can be dropped if \(G\) is required to be \(3\)-connected and to have at least \(35\) vertices.