A group divisible design (GDD) \( (v = v_1 + v_2 + \cdots + v_g, g, k; \lambda_1, \lambda_2) \) is an ordered pair \( (V, \mathcal{B}) \) where \( V \) is a \( v \)-set of symbols and \( \mathcal{B} \) is a collection of \( k \)-subsets (called blocks) of \( V \) satisfying the following properties: the \( v \)-set is divided into \( g \) groups of sizes \( v_1, v_2, \ldots, v_g \); each pair of symbols from the same group occurs in exactly \( \lambda_1 \) blocks in \( \mathcal{B} \); and each pair of symbols from different groups occurs in exactly \( \lambda_2 \) blocks in \( \mathcal{B} \). In this paper we give necessary conditions on \( m \) and \( n \) for the existence of a \( GDD(v = m+n, 2, 3; 1, 2) \), along with sufficient conditions for each \( m \leq \frac{n}{2} \). Furthermore, we introduce some construction techniques to construct some \( GDD(v = m + n, 2, 3; 1, 2) \)s when \( m > \frac{n}{2} \), namely, a \( GDD(v = 9 + 15, 2, 3; 1, 2) \) and a \( GDD(v = 25 + 33, 2, 3; 1, 2) \).

Let \( D \) be a directed graph. An anti-directed cycle in \( D \) is a set of arcs which form a cycle in the underlying graph, but for which no two consecutive arcs form a directed path in \( D \); this cycle is called an anti-directed Hamilton cycle if it includes all vertices of \( D \). Grant [6] first showed that if \( D \) has even order \( n \), and each vertex indegree and outdegree in \( D \) is a bit more than \( \frac{2n}{3} \), then \( D \) must contain an anti-directed Hamilton cycle. More recently, Busch et al. [1] lowered the lead coefficient, by showing that there must be an anti-directed Hamilton cycle if all indegrees and outdegrees are greater than \( \frac{9n}{16} \), and conjectured that such a cycle must exist if all indegrees and outdegrees are greater than \( \frac{n}{2} \). We prove that conjecture holds for all directed graphs of even order less than 20, and are thus able to extend the above result to show that any digraph \( D \) of even order \( n \) will have an anti-directed Hamilton cycle if all indegrees and outdegrees are greater than \( \frac{11n}{20} \).

Let \( G \) be a \((p,q)\)-graph in which the edges are labeled \( k, k+1, \ldots, k+q-1 \), where \( k \geq 0 \). The vertex sum for a vertex \( v \) is the sum of the labels of the incident edges at \( v \). If the vertex sums are constant, modulo \( p \), then \( G \) is said to be \( k \)-edge-magic. In this paper, we investigate some classes of cubic graphs which are \( k \)-edge-magic. We also provide a counterexample to a conjecture that any cubic graph of order \( p \equiv 2 \pmod{4} \) is \( k \)-edge-magic for all \( k \).

In this paper, we obtain a new set of conditions which are necessary for the existence of balanced arrays of strength eight with two levels by making use of the positive semi-definiteness of the matrix of moments. We also demonstrate, using illustrative examples, that the maximum number of constraints derived using these results are better than those obtained earlier.

A set \( D \subseteq V(G) \) is a dominating set of a graph \( G \) if every vertex of \( G \) not in \( D \) is adjacent to at least one vertex in \( D \). A minimum dominating set of \( G \), also called a \( \gamma(G) \)-set, is a dominating set of \( G \) of minimum cardinality. For each vertex \( v \in V(G) \), we define the domination value of \( v \) to be the number of \( \gamma(G) \)-sets to which \( v \) belongs. In this paper, we find the total number of minimum dominating sets and characterize the domination values for \( P_2 \Box P_n \), and \( P_2 \Box C_n \).

Let \( G \) be the one-point union of two cycles and suppose \( G \) has \( n \) edges. We show via various graph labelings that there exists a cyclic \( G \)-decomposition of \( K_{2nt+1} \) for every positive integer \( t \).

Decompositions of complete or near-complete graphs into spanning trees have been widely studied, but usually in the homogeneous case, where all component trees are isomorphic. A spanning tree decomposition \( \mathcal{T} = (T_1, \ldots, T_n) \) of such a graph is purely heterogeneous if no two trees \( T_i \) are isomorphic. We show existence of such decompositions with the maximum degree condition \( \Delta(T_i) = i+1 \) for each \( i \in [1..n] \), for every largest possible graph of odd order, and every even order graph which is the complement of a spanning tree satisfying a necessary maximum degree condition.

Let \( G \) be a simple graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). A labeling \( f : V(G) \to \mathbb{Z}_2 \) induces a partial edge labeling \( f^* : E(G) \to \mathbb{Z}_2 \) defined by \( f^*(uv) = f(u) \) if and only if \( f(u) = f(v) \). For \( i \in \mathbb{Z}_2 \), let \( V_f(i) = \{v \in V(G) : f(v) = i\} \) and \( e_f(i) = |\{e \in E(G) : f^*(e) = i\}| \). A labeling \( f \) is called a friendly labeling if \( |V_f(0) – V_f(1)| \leq 1 \). The \( BI(G) \), the balance index set of \( G \), is defined as \( \{|e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is friendly}\} \). This paper focuses on the balance index sets of generalized book and ear expansion graphs.

Figueroa-Centeno, Ichishima, and Muntaner-Batle [3, 4] proved some results on felicitous graphs and raised the following conjectures:

1. The one-point union of \( m \) copies of \( C_n \) is felicitous if and only if \( mn \equiv 2 \pmod{4} \).
2. \( mC_n \) is felicitous if and only if \( mn \not\equiv 2 \pmod{4} \).

In this paper, the conjectures are partially settled by proving the following results:

1. For any odd positive integers \( m \) and \( n \), the one-point union of \( m \) copies of \( C_n \) is felicitous if \( mn \equiv 1, 3 \).
2. For any positive integer \( m \), the one-point union of \( m \) copies of \( C_4 \) is felicitous.
3. For any two odd positive integers \( m \) and \( n \), \( mC_n \) is felicitous if \( mn \equiv 1, 3 \pmod{4} \).
4. For any positive integer \( m \), \( mC_4 \) is felicitous.

In this paper, we characterize the graphs \( G \) and \( H \) for which the Cartesian product \( G \Box H \) is a divisor graph. We show that divisor graphs form a proper subclass of perfect graphs. Additionally, we prove that cycle permutation graphs of order at least 8 are divisor graphs if and only if they are perfect. Some results concerning amalgamation operations for obtaining new divisor graphs from old ones are presented. We view block graphs as vertex amalgams.