In this paper, we use a genetic algorithm and direct a hill-climbing algorithm in choosing differences to generate solutions for difference triangle sets. The combined use of the two algorithms optimized the hill-climbing method and produced new improved upper bounds for difference triangle sets.

The covering problem in the \( n \)-dimensional \( q \)-ary Hamming space consists of the determination of the minimal cardinality \( K_q(n, R) \) of an \( R \)-covering code. It is known that the sphere covering bound can be improved by considering decompositions of the underlying space, leading to integer programming problems. We describe the method in an elementary way and derive about 50 new computational and theoretical records for lower bounds on \( K_q(n, R) \).

For any graph \( G = (V, E) \), \( D \subseteq V \) is a global dominating set if \( D \) dominates both \( G \) and its complement \( \overline{G} \). The global domination number \( \gamma_g(G) \) of a graph \( G \) is the fewest number of vertices required of a global dominating set. In general,

\[
\max\{\gamma(G), \gamma(\overline{G})\} \leq \gamma_g(G) \leq \gamma(G) + \gamma(\overline{G}),
\]

where \( \gamma(G) \) and \( \gamma(\overline{G}) \) are the respective domination numbers of \( G \) and \( \overline{G} \). We show that when \( G \) is a planar graph,

\[
\gamma_g(G) \leq \max\{\gamma(G) + 1, 4\}.
\]

Given an acyclic digraph \( D \), we seek a smallest sized tournament \( T \) having \( D \) as a minimum feedback arc set. The reversing number of a digraph is defined to be

\[
r(D) = |V(T)| – |V(D)|.
\]

We use integer programming methods to obtain new results for the reversing number where \( D \) is a power of a directed Hamiltonian path. As a result, we establish that known reversing numbers for certain classes of tournaments actually suffice for a larger class of digraphs.

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.