A mutation of a vertex-magic total labeling of a graph \( G \) is a swap of some set of edges incident on one vertex of \( G \) with some set of edges incident with another vertex where the labels on the two sets have the same sum. Mutation has previously been seen to be a useful method for producing new labelings from old. In this paper, we study mutations which mutate labelings of regular graphs into labelings of other regular graphs. We present results of extensive computations which confirm how prolific this procedure is. These computations add weight to MacDougall’s conjecture that all non-trivial regular graphs are vertex-magic.

A Langford-type \( m \)-tuple difference set of order \( t \) and defect \( d \) is a set of \( t \) \( m \)-tuples \( \{(d_{i,1}, d_{i,2}, \ldots, d_{i,m}) \mid i = 1, 2, \ldots, t\} \) such that \( d_{i,1} + d_{i,2} + \cdots + d_{i,m} = 0 \) for \( 1 \leq i \leq t \) and \( \{|d_{i,j}| \mid 1 \leq i \leq t, 1 \leq j \leq m\} = \{d, d+1, \ldots, d+mt-1\} \). In this paper, we give necessary and sufficient conditions on \( t \) and \( d \) for the existence of a Langford-type \( m \)-tuple difference set of order \( t \) and defect \( d \) when \( m \equiv 0, 2 \pmod{4} \). In the case that \( m \equiv 1, 3 \pmod{4} \), we provide sufficient conditions for the existence of a Langford-type \( m \)-tuple difference set of order \( t \) and defect \( d \) when \( d \) is at most about \( t/2 \). Using these results, we obtain cyclic \( m \)-cycle systems of the circulant graph \( \langle{d, d+1, \ldots, d+mt-1}\rangle_n \) for all \( n \geq 2(d+mt)-1 \) with \( d \) and \( t \) satisfying certain conditions.

We give a computer-assisted proof of the fact that \( R(K_5 – P_3, K_5) = 25 \). This solves one of the three remaining open cases in Hendry’s table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no \( (K_5 – P_3, K_5) \)-good graphs containing a \( K_4 \) on 23 or 24 vertices, where a graph \( F \) is \( (G, H) \)-good if \( F \) does not contain \( G \) and the complement of \( F \) does not contain \( H \). The unique \( (K_5 – P_3, K_5) \)-good graph containing a \( K_4 \) on 22 vertices is presented.

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.