A directed covering design, \( DC(v, k, \lambda) \), is a \( (v, k, 2\lambda) \) covering design in which the blocks are regarded as ordered \( k \)-tuples and in which each ordered pair of elements occurs in at least \( \lambda \) blocks. Let \( DE(v, k, \lambda) \) denote the minimum number of blocks in a \( DC(v, k, \lambda) \). In this paper, the values of the function \( DE(v, k, \lambda) \) are determined for all odd integers \( v \geq 5 \) and \( \lambda \) odd, with the exception of \( (v, \lambda) = (53, 1), (63, 1), (73, 1), (83, 1) \). Further, we provide an example of a covering design that cannot be directed.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). For a labeling \( f: V(G) \to A = \{0, 1\} \), define a partial edge labeling \( f^*: E(G) \to A \) such that, for each edge \( xy \in E(G) \),\(f^*(xy) = f(x) \quad \text{if and only if} \quad f(x) = f(y).\) For \( i \in A \), let \(\text{v}_f(i) = |\{ v \in V(G) : f(v) = i \}|\) and \(\text{e}_{f^*}(i) = |\{ e \in E(G) : f^*(e) = i \}|.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(
|\text{v}_f(0) – \text{v}_f(1)| \leq 1.\) If a friendly labeling \( f \) induces a partial labeling \( f^* \) such that \(|\text{e}_{f^*}(0) – \text{e}_{f^*}(1)| \leq 1,\)then \( G \) is said to be balanced. In this paper, a necessary and sufficient condition for balanced graphs is established. Using this result, the balancedness of several families of graphs is also proven.

Expanding upon a comment by P. A. Leonard [9], we exhibit \(\mathbb{Z}\)-cyclic patterned-starter based whist tournaments for \(q^2\) players, where \(g = 4k + 3\) is prime; the cases \(3 < q < 200\) are included herein, with data for \(200 < q < 5,000\) available electronically.

Let \( G \) be a \( (p, q) \)-graph and \( k \geq 0 \). A graph \( G \) is said to be k-edge-graceful if the edges can be labeled by \( k, k+1, \dots, k+q-1 \) so that the vertex sums are distinct, modulo \( p \). We denote the set of all \( k \) such that \( G \) is \( k \)-edge graceful by \( \text{egS}(G) \). The set is called the \textbf{edge-graceful spectrum} of \( G \). In this paper, we are concerned with the problem of exhibiting sets of natural numbers which are the edge-graceful spectra of the cylinder \( C_{n} \times P_{m} \), for certain values of \( n \) and \( m \).

The judgment aggregation problem is an extension of the group decision-making problem, wherein each voter votes on a set of propositions which may be logically interrelated (such as \( p \), \( p \to q \), and \( q \)). The simple majority rule can yield an inconsistent set of results, so more complicated rules must be developed. Here, the problem is cast in terms of matroids, and the Greedy Algorithm is modified to obtain a “best” result. An NP-completeness result is also presented for this particular formulation of the problem.

Let \( G \) be a connected simple \( (p, q) \)-graph and \( k \) a non-negative integer. The graph \( G \) is said to be \( k \)-edge-graceful if the edges can be labeled with \( k, k+1, \dots, k+q-1 \) so that the vertex sums are distinct modulo \( p \). The set of all \( k \) where \( G \) is \( k \)-edge-graceful is called the edge-graceful spectrum of \( G \). In 2004, Lee, Cheng, and Wang analyzed the edge-graceful spectra of certain connected bicyclic graphs, leaving some cases as open problems. Here, we determine the edge-graceful spectra of all connected bicyclic graphs without pendant.

Let \( G \) be a connected graph with vertex set \( V(G) \) and edge set \( E(G) \). A (defensive) alliance in \( G \) is a subset \( S \) of \( V(G) \) such that for every vertex \( v \in S \),\(|N[v] \cap S| \geq |N(v) \cap (V(G) – S)|.\) The alliance partition number, \( \psi_a(G) \), was defined (and further studied in [11]) to be the maximum number of sets in a partition of \( V(G) \) such that each set is a (defensive) alliance. Similarly, \( \psi_g(G) \) is the maximum number of sets in a partition of \( V(G) \) such that each set is a global alliance, i.e., each set is an alliance and a dominating set. In this paper, we give bounds for the global alliance partition number in terms of the minimum degree, which gives exactly two values for \( \psi_g(G) \) in trees. We concentrate on conditions that classify trees to have \( \psi_g(G) = i \) (\( i = 1, 2 \)), presenting a characterization for binary trees.

E. Stickel proposed a variation of the Diffie-Hellman key exchange scheme based on non-abelian groups, claiming that the underlying problem is more secure than the traditional discrete logarithm problem in cyclic groups. We show that the proposed scheme does not provide a higher level of security in comparison to the traditional Diffie-Hellman scheme.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A = \{0, 1\} \). A labeling \( f: V(G) \to A \) induces a partial edge labeling \( f^*: E(G) \to A \) defined by \(f^*(xy) = f(x), \text{ if and only if } f(x) = f(y),\) for each edge \( xy \in E(G) \). For \( i \in A \), let \(v_f(i) = \text{card}\{ v \in V(G) : f(v) = i \}\) and \(e_f^*(i) = \text{card}\{ e \in E(G) : f^*(e) = i \}.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(|v_f(0) – v_f(1)| \leq 1.\) If \(|e_f(0) – e_f(1)| \leq 1,\)then \( G \) is said to be \textbf{\emph{balanced}}. The \textbf{\emph{balance index set}} of the graph \( G \), \( BI(G) \), is defined as \(BI(G) = \{ |e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is friendly} \}.\)Results parallel to the concept of friendly index sets are pr

An orthogonal double cover (ODC) of the complete graph \( K_n \) by a graph \( G \) is a collection \( \mathcal{G} = \{G_i \mid i=1,2,\dots,n\} \) of spanning subgraphs of \( K_n \), all isomorphic to \( G \), with the property that every edge of \( K_n \) belongs to exactly two members of \( \mathcal{G} \) and any two distinct members of \( \mathcal{G} \) share exactly one edge.

A lobster of diameter five is a tree arising from a double star by attaching any number of pendant vertices to each of its vertices of degree one. We show that for any double star \( R(p, q) \) there exists an ODC of \( K_n \) by all lobsters of diameter five (with finitely many possible exceptions) arising from \( R(p, q) \).