In \([B]\), Bondy conjectured that if \(G\) is a \(2\)-edge-connected simple graph with \(n\) vertices, then \(G\) admits a cycle cover with at most \((2n-1)/{3}\) cycles. In this note, we show that if \(G\) is a \(2\)-edge-connected simple graph with \(n\) vertices and without subdivisions of \(K_4\), then \(G\) has a cycle cover with at most \((2n-2)/{3}\) cycles and we characterize all the extremal graphs. We also show that if \(G\) is \(2\)-edge-connected and has no subdivision of \(K_4\), then \(G\) is mod \((2k+1)\)-orientable for any integer \(k \geq 1\).

A construction of rectangular designs from Bhaskar Rao designs is described. As special cases some series of rectangular designs are obtained.

A graph \(G\) is called \((d,d+1)\)-graph if the degree of every vertex of \(G\) is either \(d\) or \(d+1\). In this paper, the following results are proved:
A \((d,d+1)\)-graph \(G\) of order \(2n\) with no \(1\)-factor and no odd component, satisfies \(|V(G)| \geq 3d+4\);A \((d,d+1)\)-graph \(G\) of order \(2n\) with \(d(G) \geq n\), contains at least \([(n+2)/{3}] + (d-n)\) edge disjoint \(1\)-factors.These results generalize the theorems due to W. D. Wallis, A. I. W. Hilton and C. Q. Zhang.

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.