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.