In this paper, we obtain some combinatorial inequalities involving the parameters of a balanced array (B-array) \(T\) of strength four and with two levels. We discuss the usefulness of these inequalities in obtaining an upper bound for the number of constraints of \(T\), and briefly describe the importance of these arrays in the design of experiments as well as in combinatorics.

We call a partition \(\mu = (\mu_1, \ldots, \mu_k)\) of \(m\), \(m \leq n\), a constrained induced partition (cip) from a partition \(\lambda = (\lambda_1, \ldots, \lambda_r)\) of \(n\) if \(\mu_i \leq \lambda_i\) for \(i = 1, 2, \ldots, k\). In this paper, we study the set of cips (Sections 1-2), determine cips of size \(p\) (Section 4), and give a formula for the number of total subsequences with fixed size chosen from a given multiset such that the multiplicity of each digit in a subsequence is less than or equal to the multiplicity of this digit in the given multiset.

Let \(n \geq 2\) be an arbitrary integer. We show that for any two asymmetric digraphs \(D\) and \(F\) with \(m\)-\(\text{rad} F \geq \max\{4, n+1\}\), there exists an asymmetric digraph \(H\) such that \(m_M(H) \cong D\), \(m_P(H) \cong F\), and \(md(D, F) = n\).
Furthermore, if \(K\) is a nonempty asymmetric digraph isomorphic to an induced subdigraph of both \(D\) and \(F\), then there exists a strong asymmetric digraph \(H\) such that
\(m_M(H) \cong D\), \(m_P(H) \cong F\), and \(m_M(H) \cap m_P(H) \cong K\) if \(m\)-\(\text{rad}_{H_0}F \geq 4\), where \(H_0\) is a digraph obtained from \(D\) and \(F\) by identifying vertices similar to those in \(K\).

This paper addresses the following questions. In any graph \(G\) with at least \(\alpha\binom{n}{2}\) edges, how large of an induced subgraph \(H\) can we guarantee the existence of with minimum degree \(\delta(H) \geq \lfloor\alpha|V(H)|\rfloor\)? In any graph \(G\) with at least \(\alpha\binom{n}{2} – f(n)\) edges, where \(f(n)\) is an increasing function of \(n\), how large of an induced subgraph \(H\) can we guarantee the existence of containing at least \(\alpha\binom{|V(H)|}{2}\) edges? In any graph \(G\) with at least \(\alpha n^2\) edges, how large of an induced subgraph \(H\) can we guarantee the existence of with at least \(\alpha|V(H)|^2 + \Omega(n)\) edges? For \(\alpha = 1 – \frac{1}{r}\), for \(r = 2, 3, \ldots\), the answer is zero since if \(G\) is a complete \(r\)-partite graph, no subgraph \(H\) of \(G\) has more than \(\alpha|V(H)|^2\) edges. However, we show that for all admissible \(\alpha\) except these, the answer is \(\Omega(n)\). In any graph \(G\) with minimum degree \(\delta(G) \geq \alpha n – f(n)\), where \(f(n) = o(n)\), how large of an induced subgraph \(H\) can we guarantee the existence of with minimum degree \(\delta(H) \geq \Omega|V(H)|\)?

From any projective plane \(\Pi\) of even order \(n\) with an oval (\((n+2)\)-arc), a Hadamard \(3\)-design on \(n^2\) points can be defined using a well-known construction. If \(\Pi\) is Desarguesian with \(n = 2^m\) and the oval is regular (a conic plus nucleus) then
it is shown that the binary code of the Hadamard \(3\)-design contains a copy of the first-order Reed-Muller code of length \(2^{2m}\).

The \emph{geodetic cover} of a graph \(G = (V, E)\) is a set \(C \subseteq V\) such that
any vertex not in \(C\) is on some shortest path between two vertices of \(C\). A minimum geodetic cover is called a \emph{geodetic basis}, and the size of a geodetic basis is called the \emph{geodetic number}. Recently, Harary, Loukakis, and Tsouros announced that finding the geodetic number of a graph is NP-Complete. In this paper, we prove a stronger result, namely that the problem remains NP-Complete even when restricted to chordal graphs. We also show that the problem of computing the geodetic number for split graphs is solvable in polynomial time.

We exhibit a self-conjugate self-orthogonal diagonal Latin square of order \(25\).

We consider the problem of scheduling \(n\) independent tasks
on a single processor with generalized due dates. The due dates
are given according to positions at which jobs are completed,
rather than specified by the jobs.
We show that the following problems are NP-Complete,
\(1|\text{prec}, p_j = 1|\sum w_jU_j\),
\(1|\text{chain}, p_j = 1|\sum w_jU_j\),
\(1|\text{prec}, p_j = 1|\sum w_jT_j\), and
\(1|\text{chain}, p_j = 1|\sum w_j T_j\).
With the removal of precedence constraints, we prove that
the two problems,
\(1|p_j = 1|\sum w_jU_j\) and
\(1|p_j = 1|\sum w_j T_j\),
are polynomially solvable.

The paper studies linear block codes and syndrome functions built by the greedy loop transversal algorithm. The syndrome functions in the binary white-noise case are generalizations of the logarithm, exhibiting curious fractal properties. The codes in the binary white-noise case coincide with lexicodes; their dimensions are listed for channel lengths up to sixty, and up to three hundred for double errors. In the ternary double-error case, record-breaking codes of lengths \(43\) to \(68\) are constructed.