The intersection problem for a pair of transitive triple systems (or $2-(v,3,1)$ directed designs) was solved by Lindner and Wallis, and independently by H.L. Fu, in 1982-1983. In this paper, we determine the intersection problem for a pair of $2-(v,4,1)$ directed designs.

Let $H_a^1=\{v:v\ge a,v\equiv 0,1(\mathrm{mod}a)\}$. It is well known that such sets are PBD-closed. Finite bases are found for these sets for $a=6,7,$ and $8$. At the same time, we improve the result of Mullin in [6] about finite bases of $H^a=\{v:v\ge a+1,v\equiv 1(\mathrm{mod}a)\}$ for $a=5$ and $6$.

Let $G$ be a $2$-edge-connected graph and $v$ be a vertex of $G$, and $F\subset F^{\prime }\subset E(v)$ such that $1\le |F|$ and $|F|+2=|F^{\prime }|\le d(v)–1$. Then there is a subset $F^{\ast }$ such that $F\subset F^{\ast }\subset F^{\prime }$ (here, $|F^{\ast }|=|F|+1$), and the graph obtained from $G$ by splitting the edges of $F^{\ast }$ away from $v$ remains $2$-edge-connected unless $v$ is a cut-vertex of $G$. This generalizes a very useful Vertex-Splitting Lemma of Fleischner.
Let $\mathcal{C}$ be a circuit cover of a bridge-less graph $G$. The depth of $\mathcal{C}$ is the smallest integer $k$ such that every vertex of $G$ is contained in at most $k$ circuits of $\mathcal{C}$. It is conjectured by L. Pyber that every bridge-less graph $G$ has a circuit cover $\mathcal{C}$ such that the depth of $\mathcal{C}$ is at most $\mathrm{\Delta }(G)$. In this paper, we prove that

1. every bridge-less graph $G$ has a circuit cover $\mathcal{C}$ such that the depth of $\mathcal{C}$ is at most $\mathrm{\Delta }(G)+2$ and
2. if a bridge-less graph $G$ admits a nowhere-zero $4$-flow or contains no subdivision of the Petersen graph, then $G$ has a circuit cover $\mathcal{C}$ such that the depth of $\mathcal{C}$ is at most $2\lceil 2\mathrm{\Delta }(G)/3\rceil$.

Let $m\ge 1$ be an integer and let $G$ be a graph of order $n$. A set $\mathcal{D}$ of vertices of $G$ is an $m$-dominating set of $G$ if every vertex of $V(G)–\mathcal{D}$ is within distance $m$ from some vertex of $\mathcal{D}$. An independent set of vertices of $G$ is a set of vertices of $G$ whose elements are pairwise nonadjacent. The minimum cardinality among all independent $m$-dominating sets of $G$ is called the independent $m$-domination number and is denoted by $id(m,G)$. We show that if $G$ is a connected graph of order $n\ge m+1$, then $id(m,G)\le (n+m+1-2\sqrt{n}/m),$ and this bound is sharp.

Several theorems about Hamiltonian, pan-cyclic and other properties of locally semi-complete digraphs are obtained in this paper.

In the $n$-dimensional hypercube, an $n$-snake is a simple path with no chords, while an $n$-coil is a simple cycle without chords. There has been much interest in determining the length of a maximum $n$-snake and a maximum $n$-coil. Only upper and lower bounds for these maximum lengths are known for arbitrary $n$. Computationally, the problem of finding maximum $n$-snakes and $n$-coils suffers from combinatorial explosion, in that the size of the solution space which must be searched grows very rapidly as $n$ increases. Previously, the maximum lengths of $n$-snakes and $n$-coils have been established only for $n\le 7$and $n\le 6$, respectively. In this paper, we report on a coil searching computer program which established that $48$ is the maximum length of a coil in the hypercube of dimension $7$.

The complements of the perfect dominating sets of the $n$-cube, for $n\le 8$, are characterized as well as some outstanding vertex-spanning edge-partitions of them involving the Fano plane, as a contribution to the study of distance-preserving regular subgraphs of hypercubes.

A graph is said to be in $L_1$ if $\mathrm{deg}(u)+\mathrm{deg}(v)\ge |N(u)\cup N(w)\cup N(v)|–1$ for each induced path $uwv$ of order three. We prove that a $2$-connected graph $G$ in $L_1$ of diameter two is hamiltonian, or $K_{d,d+1}\subset G\subset K_d+(d+1)K_1$ for some $d\ge 2$. This theorem generalizes a couple of known sufficient conditions for a graph to be hamiltonian. We also discuss the relation between this theorem and several other degree conditions for hamiltonicity.

On the basis of circuit uniqueness, the concept of strong circuit uniqueness is introduced, and some graphs with the property of strong circuit uniqueness are identified. The results are then used to prove successfully the circuit uniqueness of the graphs $K_m\cup K_n$ and $K_{m,n}$. This represents an improvement on the previous papers on the same subject.

Several criteria have been proposed as desirable for binary cryptographic functions. Three important ones are balance, correlation-immunity, and higher order strict avalanche criterion. Lloyd [7] has shown that there are no balanced, uncorrelated functions which satisfy the strict avalanche criterion of order $n-2$. In this note, we give a short proof of this result using elementary combinatorial arguments. The proof relies on the solution of a recurrence relation that seems to be of interest in its own right.