It is shown that the integrity of the \(n\)-dimensional cube is \(O(2^n \log n/\sqrt{n})\).

We discuss the learning problem in a two-layer neural network. The problem is reduced to a system of linear inequalities, and the solvability of the system is discussed.

We show how to generate \(k \times n\) Latin rectangles uniformly at random in expected time \(O(nk^3)\), provided \(k = o(n^{1/3})\). The algorithm uses a switching process similar to that recently used by us to uniformly generate random graphs with given degree sequences.

For any integers \(r\) and \(n\), \(2 < r < n-1\), it is proved that there exists an order \(n\) regular graph of degree \(r\) whose amida number is \(r + 1\).

An \(h\)-cluster in a graph is a set of \(h\) vertices which maximizes the number of edges in the graph induced by these vertices. We show that the connected \(h\)-cluster problem is NP-complete on planar graphs.

Lee conjectures that for any \(k > 1\), a \((n,nk)\)-multigraph decomposable into \(k\) Hamiltonian cycles is edge-graceful if \(n\) is odd. We investigate the edge-gracefulness of a special class of regular multigraphs and show that the conjecture is true for this class of multigraphs.

A balanced incomplete block design \(B[k, \alpha; v]\) is said to be a nested design if one can add a point to each block in the design and so obtain a block design \(B[k + 1, \beta; v]\). Stinson (1985) and Colbourn and Colbourn (1983) proved that the necessary condition for the existence of a nested \(B[3, \alpha; v]\) is also sufficient. In this paper, we investigate the case \(k = 4\) and show that the necessary condition for the existence of a nested \(B[4, \alpha; v]\), namely \(\alpha = 3\lambda\), \(\lambda(v – 1) \equiv 0 \pmod{4}\) and \(v \geq 5\), is also sufficient. To do this, we need the concept of a doubly nested design. A \(B[k, \alpha; v]\) is said to be doubly nested if the above \(B[k + 1, \beta; v]\) is also a nested design. When \(k = 3\), such a design is called a doubly nested triple system. We prove that the necessary condition for the existence of a doubly nested triple system \(B[3, \alpha; v]\), namely \(\alpha = 3\lambda\), \(\lambda(v – 1) \equiv 0 \pmod{2}\) and \(v \geq 5\), is also sufficient with the four possible exceptions \(v = 39\) and \(\alpha = 3, 9, 15, 21\).