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.

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 \( R(p, q) \).

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 an edge partial labeling \( f^*: E(G) \to A \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E(G) \). For each \( i \in A \), let \(v_f(i) = |\{v \in V(G) : f(v) = i\}|\) and \(e_f(i) = |\{e \in E(G) : f^*(e) = i\}|.\)The balance index set of \( G \), denoted \( BI(G) \), is defined as \(\{|e_f(0) – e_f(1)|: |v_f(0) – v_f(1)| \leq 1\}.\)In this paper, exact values of the balance index sets of five new families of one-point union of graphs are obtained, many of them, but not all, form arithmetic progressions.

For any \( h \in \mathbb{Z} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_h \), defined by
\[
l^+(v) = \sum_{uv \in E(G)} l(uv)\]
is a constant map. For a given graph \( G \), the set of all \( h \in \mathbb{Z}_+ \) for which \( G \) is \( h \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). In this paper, we will determine the integer-magic spectra of trees of diameter five.

A graceful labeling of a directed graph \( D \) with \( e \) edges is a one-to-one map \( \theta: V(D) \to \{0, 1, \dots, e\} \) such that \( \theta(y) – \theta(x) \mod (e + 1) \) is distinct for each \( (x, y) \in E(D) \). This paper summarizes previously known results on graceful directed graphs and presents some new results on directed paths, stars, wheels, and umbrellas.

For an integer \( l > 1 \), the \( l \)-edge-connectivity of a graph \( G \) with \( |V(G)| \geq l \), denoted by \( \lambda_l(G) \), is the smallest number of edges whose removal results in a graph with \( l \) components. In this paper, we study lower bounds of \( \lambda_l(G) \) and optimal graphs that reach the lower bounds. Former results by Boesch and Chen are extended.

We also present in this paper an optimal model of interconnection network \( G \) with a given \( \lambda_l(G) \) such that \( \lambda_2(G) \) is maximized while \( |E(G)| \) is minimized.

Given an abelian group \( A \), a graph \( G = (V, E) \) is said to have a distance two magic labeling in \( A \) if there exists a labeling \( l: E(G) \to A – \{0\} \) such that the induced vertex labeling \( l^*: V(G) \to A \) defined by

\[l^*(v) = \sum_{c \in E(v)} l(e)\]

is a constant map, where \( E(v) = \{e \in E(G) : d(v,e) < 2\} \). The set of all \( h \in \mathbb{Z}_+ \), for which \( G \) has a distance two magic labeling in \( \mathbb{Z}_h \), is called the distance two magic spectrum of \( G \) and is denoted by \( \Delta M(G) \). In this paper, the distance two magic spectra of certain classes of graphs will be determined.

In this paper, we derive some necessary existence conditions for a bi-level balanced array (B-array) with strength \( t = 5 \). We then describe how these existence conditions can be used to obtain an upper bound on the number of constraints of these arrays, and give some illustrative examples to this effect.

Let \( G = (V, E) \) be a graph with a vertex labeling \( f: V \to \mathbb{Z}_2 \) that induces an edge labeling \( f^*: E \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) + f(y) \). For each \( i \in \mathbb{Z}_2 \), let \(
v_f(i) = \text{card}\{v \in V: f(v) = i\}\) and \(e_f(i) = \text{card}\{e \in E: f^*(e) = i\}.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(\lvert v_f(0) – v_f(1) \rvert \leq 1.\) The friendly index set of \( G \) is defined as \(\{\lvert e_f(1) – e_f(0) \rvert : \text{the vertex labeling } f \text{ is friendly}\}.\)
In this paper, we determine the friendly index sets of generalized books.

Given 2 triangles in a plane over a field \( F \) which are in perspective from a vertex \( V \), the resulting Desargues line or axis \( l \) may or may not be on \( V \). To avoid degenerate cases, we assume that the union of the vertices of the 2 triangles is a set of six points with no three collinear. Our work then provides a detailed analysis of situations when \( V \) is on \( l \) for any \( F \), finite or infinite.

We give constructive and combinatorial proofs to decide why certain families of slightly irregular graphs have no planar representation and why certain families have such planar representations. Several non-existence results for infinite families as well as for specific graphs are given. For example, the nonexistence of the graphs with \( n = 11 \) and degree sequence \( (5, 5, 5, \ldots, 4) \) and \( n = 13 \) and degree sequence \( (6, 5, 5, \ldots, 5) \) are shown.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). Let \( A = \{0, 1\} \). A labeling \( f: V(G) \to A \) induces a partial edge labeling \( f^*: E(G) \to A \) defined by \(f^*(xy) = f(x) \quad \text{if and only if } f(x) = f(y),\) 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 \(\lvert v_f(0) – v_f(1) \rvert \leq 1.\)If \(\lvert e_{f^*}(0) – e_{f^*}(1) \rvert \leq 1,\) then \( G \) is said to be \(\textbf{balanced}\). The balancedness of the Cartesian product and composition of graphs is studied in [19]. We provide some new families of balanced graphs using other constructions.