Bondy conjectures that if $G$ is a $2$-edge-connected simple graph with $n$ vertices, then at most $(2n-1)/3$ cycles in $G$ will cover $G$. In this note, we show that if $G$ is a plane triangulation with $n\ge 6$ vertices, then at most $(2n-3)/3$ cycles in $G$ will cover $G$.

An affine (respectively projective) failed design $D$, denoted by $\text{AFD}(q)$ (respectively $\text{PFD}(q)$) is a configuration of $v=q^2$ points, $b=q^2+q+1$ blocks and block size $k=q$ (respectively $v=q^2+q+1$ points, $b=q^2+q+2$ blocks and block size $k=q+1$) such that every pair of points occurs in at least one block of $D$ and $D$ is minimal, that is, $D$ has no block whose deletion gives an affine plane (respectively a projective plane) of order $q$. These configurations were studied by Mendelsohn and Assaf and they conjectured that an $\text{AFD}(q)$ exists if an affine plane of order $q$ exists and a $\text{PFD}(q)$ never exists. In this paper, it is shown that an $\text{AFD}(5)$ does not exist and, therefore, the first conjecture is false in general, $\text{AFD}(q^2)$ exists if $q$ is a prime power and the second conjecture is true, that is, $\text{PFD}(q)$ never exists.

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 $k\ge 1$.

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 $|V(G)|\ge 3d+4$;A $(d,d+1)$-graph $G$ of order $2n$ with $d(G)\ge n$, 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 $O(n{k}^3)$, 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 $r+1$.