We prove that the complete graph \( K_v \) can be decomposed into rhombicuboctahedra if and only if \( v \equiv 1 \) or \( 33 \pmod{96} \).

A starter in an odd order abelian group \( G \) is a set of unordered pairs \( S = \{\{s_i, t_i\} : 1 \leq i \leq \frac{|G| – 1}{2}\} \), for which \( \{s_i\} \cup \{t_i\} = G \setminus \{0\} \) and \( \{\pm(s_i – t_i)\} = G \setminus \{0\} \). If \( s_i + t_i = s_j + t_j \) holds only for \( i = j \), then the starter is called a strong starter. Only cyclic groups are considered in this work, where starters and strong starters up to order \( 35 \) and \( 37 \), respectively, are classified using an exact cover algorithm. The results are validated by double counting.

This paper settles in the negative the following open question: Are \( V_4 \)-magic graphs necessarily \( \mathbb{Z}_4 \)-magic? For an abelian group \( A \), we examine the properties of \( A \)-magic labelings with constant weight \( 0 \), called \({zero-sum \; A -magic}\), and utilize well-known results on edge-colorings in order to construct (from \( 3 \)-regular graphs) infinite families that are \( V_4 \)-magic but not \( \mathbb{Z}_4 \)-magic. Noting that our arguments lead to connected graphs of order \( 2n \) for all \( n \geq 11 \) that are \( V_4 \)-magic and not \( \mathbb{Z}_4 \)-magic, we conclude the paper by investigating the zero-sum integer-magic spectra of graphs, including Cartesian products, and give a conjecture about the zero-sum integer-magic spectra of \( 3 \)-regular graphs.

A new technique is given for constructing a vertex-magic total labeling, and hence an edge-magic total labeling, for certain finite simple \(2\)-regular graphs. Let \( C_r \) denote the cycle of length \( r \). Let \( n \) be an odd positive integer with \( n = 2m + 1 \). Let \( k_i \) denote an integer such that \( k_i \geq 3 \), for \( i = 1, 2, \ldots, l \), and write \( nC_{k_i} \) to mean the disjoint union of \( n \) copies of \( C_{k_i} \). Let \( G \) be the disjoint union \( G \cong C_{k_1} \cup \ldots \cup C_{k_l} \). Let \( I = \{1, 2, \ldots, l\} \) and let \( J \) be any subset of \( I \). Finally, let \( G_J = \left(\bigcup_{i \in J} nC_{k_i}\right) \cup \left(\bigcup_{i \in I – J} C_{nk_i}\right) \), where all unions are disjoint unions. It is shown that if \( G \) has a vertex-magic total labeling (VMTL) with a magic constant of \( h \), then \( G_J \) has VMTLs with magic constants \( 6m(k_1 + k_2 + \ldots + k_l) + h \) and \( nh – 3m \). In particular, if \( G \) has a strong VMTL then \( G_J \) also has a strong VMTL.

The threshold dimension of a graph is the minimum number of threshold subgraphs needed to cover its edges. In this work, we present a new characterization of split-permutation graphs and prove that their threshold dimension is at most two. As a consequence, we obtain a structural characterization of threshold graphs.

In this paper, we construct inequivalent Hadamard matrices based on Yang multiplication methods for base sequences which are obtained from near normal sequences. This has been achieved by employing various Unix tools and sophisticated techniques, such as metaprogramming. In addition, we present a classification for near normal sequences of length \( 4n + 1 \), for \( n \leq 11 \) and some of these for \( n = 12, 13, 14, 15 \), taking into account previously known results. Finally, we improve several constructive lower bounds for inequivalent Hadamard matrices of large orders.

We give an upper bound on the number of edges of a graph with \( n \) vertices to be a prime cordial graph, and we improve this upper bound to fit bipartite graphs. Also, we determine all prime cordial graphs of order \( \leq 6 \).

We consider the one-color graph avoidance game. Using a high-performance computing network, we showed that the first player can win the game on \( 13 \), \( 14 \), and \( 15 \) vertices. Other related games are also discussed.

Let \( G \, \Box \, H \) denote the Cartesian product of two graphs \( G \) and \( H \). In 1994, Livingston and Stout [Constant time computation of minimum dominating sets, Congr. Numer., 105 (1994), 116-128] introduced a linear time algorithm to determine \( \gamma(G \, \Box \, P_n) \) for fixed \( G \), and claimed that \( P_n \) may be substituted with any graph from a one-parameter family, such as a cycle of length \( n \) or a complete \( t \)-ary tree of height \( n \) for fixed \( t \). We explore how the algorithm may be modified to accommodate such graphs and propose a general framework to determine \( \gamma(G \, \Box \, H) \) for any graph \( H \). Furthermore, we illustrate its use in determining the domination number of the generalized Cartesian product \( G \, \Box \, H \), as defined by Benecke and Mynhardt [Domination of Generalized Cartesian Products, preprint (2009)].

We give a solution for the intersection problem for disjoint \( 2 \)-flowers in Steiner triple systems.