Let \( \mathcal{B} \subseteq 2^{[m]} \) be an antichain of size \( |\mathcal{B}| =: n \). \( 2^{[m]} \) is ordered by inclusion. An antichain \( \mathcal{B} \) is called \( k \)-regular (\( k \in \mathbb{N} \)), if for each \( i \in [m] \) there are exactly \( k \) sets \( B_1, B_2, \ldots, B_k \in \mathcal{B} \) containing \( i \). In this case, we say that \( \mathcal{B} \) is a \( (k, m, n) \)-antichain.

Let \( m \geq 2 \) be an arbitrary natural number. In this note, we show that an \( (m-1, m, n) \)-antichain exists if and only if \( n \in [m+2, \binom{m}{2} – 2] \cup \{m, \binom{m}{2}\} \).

Let \( G = (V, E) \) be a connected graph. A subset \( S \) of \( V \) is called a degree equitable set if the degrees of any two vertices in \( S \) differ by at most one. The minimum order of a partition of \( V \) into independent degree equitable sets is called the \({degree \;equitable\; chromatic \;number}\) of \( G \) and is denoted by \( \chi_{de}(G) \). In this paper, we initiate a study of this new coloring parameter.

An avoidance problem of configurations in \( 4 \)-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erdős’ \( r \)-sparse conjecture on Steiner triple systems. A \( 4 \)-cycle system of order \( v \), \( 4CS(v) \), is said to be \( r \)-sparse if for every integer \( j \) satisfying \( 2 \leq j \leq r \) it contains no configurations consisting of \( j \) \( 4 \)-cycles whose union contains precisely \( j + 3 \) vertices. If an \( r \)-sparse \( 4CS(v) \) is also free from copies of a configuration on two \( 4 \)-cycles sharing a diagonal, called the double-diamond, we say it is strictly \( r \)-sparse. In this paper, we show that for every admissible order \( v \) there exists a strictly \( 4 \)-sparse \( 4CS(v) \). We also prove that for any positive integer \( r \geq 2 \) and sufficiently large integer \( v \), there exists a constant number \( c \) such that there exists a strictly \( r \)-sparse \( 4 \)-cycle packing of order \( v \) with \( c \cdot v^2 \) \( 4 \)-cycles.

A set of Hamilton cycles in the complete graph \( K_n \) is called a Dudeney set if every path of length two lies on exactly one of the cycles. It has been conjectured that there is a Dudeney set for every complete graph. It is known that there exists a Dudeney set for \( K_n \) when \( n \) is even, but the question is still unsettled when \( n \) is odd.

In this paper, we define a black \( 1 \)-factor in \( K_{p+1} \) for an odd prime \( p \), and show that if there exists a black \( 1 \)-factor in \( K_{p+1} \), then we can construct a Dudeney set for \( K_{p+2} \). We also show that if there is a black \( 1 \)-factor in \( K_{p+1} \), then \( 2 \) is a quadratic residue modulo \( p \). Using this result, we obtain some new Dudeney sets for \( K_n \) when \( n \) is odd.

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.