We show that the edges of a planar 3-connected graph with \(n\) vertices can be covered by at most \([(n + 1)/{2}]\) cycles. This proves a special case of a conjecture of Bondy that the edges of a 2-connected graph can be covered by at most \((2n – 1)/{3}\) cycles.

In [J. of Combinatorial Theory (B),40(1986),229-230], Fleischner proved that if \(G\) is a \(2\)-edge-connected planar graph and if \(\mathcal{C}_0 = \{C_1, \ldots, C_k\}\) is a collection of edge-disjoint cycles of \(G\), then \(G\) has a cycle double cover \(\mathcal{C}\) that contains \(\mathcal{C}_0\). In this note, we show that this holds also for graphs that do not have a subgraph contractible to \(K_5\).

Given a binary code \(C\), the set \(K\) of all vectors which leave \(C\) invariant under translation is called the kernel of \(C\). The main concern of this paper is the development of an efficient algorithm for computing the kernel of \(C\). We present such an algorithm with runtime \(O(|C| \log |C|)\), which is the best possible.

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 \(H_a^1 = \{v: v \geq a, v \equiv 0,1 \pmod{a}\}\). 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 \(H^a = \{v: v \geq a+1, v \equiv 1 \pmod{a}\}\) for \(a = 5\) and \(6\).

Let \(G\) be a \(2\)-edge-connected graph and \(v\) be a vertex of \(G\), and \(F \subset F’ \subset E(v)\) such that \(1 \leq |F|\) and \(|F| + 2 = |F’| \leq d(v) – 1\). Then there is a subset \(F^*\) such that \(F \subset F^* \subset F’\) (here, \(|F^*| = |F| + 1\)), and the graph obtained from \(G\) by splitting the edges of \(F^*\) 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 \(\Delta(G)\). 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 \(\Delta(G) + 2\) 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 \(2 \left\lceil 2\Delta(G)/3 \right\rceil\).

Let \(m \geq 1\) 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 \(n \geq m + 1\), then \(id(m,G) \leq ({n+m+1-2\sqrt{n}/{m}}),\) and this bound is sharp.

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