In this paper, we study the minimum co-operative guards problem, a variation of the art gallery problem. First, we show that the minimum number of co-operative guards required for a \(k\)-spiral polygon is at most \(N_k\), the total number of reflex vertices in the \(k\)-spiral. Then, we classify \(2\)-spirals into seven different types based on their structure. Finally, we present a minimum co-operative guard placement algorithm for general
\(2\)-spirals.

In this paper, we construct all symmetric \(27, 13, 6\) designs with a fixed-point-free automorphism of order \(3\). There are \(22\) such designs.

In this paper, the Desargues Configuration in \({P}^2(k)\), where \(k\) is a field of characteristic \( \neq 2\), is characterized combinatorially en route to define Desargues Block Designs and associate them with certain families of dihedral subgroups of \(S_6\) through the use of the outer automorphisms of \(S_6\).

Fix a positive integer \(k\). A mod \(k\)-orientation of a graph \(G\) is an assignment of edge directions to \(E(G)\) such that at each vertex \(v \in V(G)\), the number of edges directed in is congruent to the number of edges directed out
modulo \(k\). The main purpose of this note is to correct an error in [JCMCC, 9 (1991), 201-207] by showing that a connected graph \(G\) has a mod \((2p + 1)\)-orientation for any \(p \geq 1\) if and only if \(G\) is Eulerian.

We report on progress towards deciding the existence of \(2-(22, 8, 4)\) designs without assuming any automorphisms. Using computer algorithms, we have shown that in any such design every two blocks have nonempty intersection, every quadruple of points can occur in at most two blocks, and no three blocks can pairwise intersect in a single point.

A graph \(P_{n}^{2}\), \(n \geq 3\), is the graph obtained from a path \(P_{n}\) by adding edges that join all vertices \(u\) and \(v\) with \(d(u,v) = 2\). A graph \(C_{n}^{+t}\), \(n \geq 3\) and \(1 \leq t \leq n\), is formed by adding a single pendent edge to \(t\) vertices of a cycle of length \(n\). A Web graph \(W(2,n)\) is obtained by joining the pendent vertices of a Helm graph (i.e., a Wheel graph with a pendent edge at each cycle vertex) to form a cycle and then adding a single pendent edge to each vertex of this outer cycle. In this paper, we find the gracefulness of \(P_{n}^{2}\) for any \(n\), of \(C_{n}^{+t}\) for \(n \geq 3\) and \(1 \leq t \leq n\), and of \(W(2,n)\) for \(n \geq 3\). Therefore, three conjectures about labeling graphs —Grace’s, Koh’s, and Gallian’s — are confirmed.

A problem about “nine foreign journalists” from a Nordic Mathematical Olympiad is used as the starting point for a discussion of a class of extremal problems involving hypergraphs.
Specifically, the problem is to find a sharp lower bound for the maximum degree of the hypergraph in terms of the number of (hyper)edges and their cardinalities.

In [Discrete Math. 111 (1993), 113-123], the \(c\)-th order edge toughness of a graph \(G\) is defined as
\[
\tau_c(G) = \min_{\substack{X \subseteq E(G), \&\omega(G – X) > c }} \left\{\frac{|X|}{\omega(G – X) – c}\right\},
\]
for any \(1 \leq c \leq |V(G)| – 1\). It is proved that \(\tau_c(G) \geq k\) if and only if \(G\) has \(k\) edge-disjoint spanning forests with exactly \(c\) components, and that for a given graph \(G\) with \(s = |E(G)|/(|V(G)| – c)\) and \(1 \leq c \leq |E(G)|\),
\(\tau_c(G) = s\) if and only if \(|E(H)| \leq s(|V(H)| – 1)\) for any subgraph \(H\) of \(G\). In this note, we shall present short proofs of the abovementioned theorems and shall indicate that these results can be extended to matroids.

In a group channel, codes correcting and detecting arbitrary patterns of errors (not necessarily “white noise”) are described metrically. This yields sphere-packing and Gilbert bounds on the sizes of all and of maximal codes, respectively. The loop transversal approach builds linear codes correcting arbitrary error patterns. In the binary case, the greedy loop transversal algorithm builds lexicodes.

A \(\lambda\)-design on \(v\) points is a family of \(v\) subsets (blocks) of a \(v\)-set such that any two distinct blocks intersect in \(\lambda\) points and not all blocks have the same cardinality.Ryser’s and Woodall’s \(\lambda\)-design conjecture states that each \(\lambda\)-design can be obtained from a symmetric design by complementing with respect to a fixed block. In a recent paper, we proved this conjecture for \(v = p+1, 2p+1, 3p+1\), where \(p\) is prime, and remarked that similar methods might work for \(v = 4p+1\). In the present paper, we prove the conjecture for \(\lambda\)-designs having replication numbers \(r\) and \(r^*\) such that \((r-1, r^*-1) = 4\) and, as a consequence, the \(\lambda\)-design conjecture is proved for \(v = 4p+1\), where \(p\) is prime.

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.

Suppose that a finite group \(G\) acts on two sets \(X\) and \(Y\), and that \(FX\) and \(FY\) are the natural permutation modules for a field \(F\). We examine conditions which imply that \(FX\) can be embedded in \(FY\), in other words that \((\ast)\): There is
an injective \(G\)-map \( FX \rightarrow FY\). For primitive groups we show that \((\ast)\)
holds if the stabilizer of a point in \(Y\) has a `maximally overlapping’ orbit on \(X\). For groups of rank three, we show that \((\ast)\) holds unless a specific divisibility condition on the eigenvalues of an orbital matrix of \(G\) is satisfied. Both results are obtained by constructing suitable incidence geometries.

A Latin square \((S, \ast)\) is said to be \((3,2,1)\)-conjugate-orthogonal if \(x \ast y = z \ast w\), \(x \ast_{321} y\), \(z \ast_{321} w\) imply \(x = z\) and \(y = w\), for all \(x, y, z, w \in S\), where \(x_3 \ast_{321} x_2 = x_1\) if and only if \(x_1 \ast x_2 = x_3\). Such a Latin square is said to be \emph{holey}(\((3,2,1)\)-HCOLS for short) if it has disjoint and spanning holes corresponding to missing sub-Latin squares.
Let \((3,2,1)\)-HCOLS\((h^n)\) denote a \((3,2,1)\)-HCOLS of order \(hn\) with \(n\) holes of equal size \(h\). We show that, for any \(h \geq 1\), a \((3,2,1)\)-HCOLS\((h^n)\) exists if and only if \(n \geq 4\), except \((n,h) = (6,1)\) and except possibly \((n,h) = (6,13)\). In addition, we show that a \((3,2,1)\)-HCOLS with \(n\) holes of size \(2\)
and one hole of size \(3\) exists if and only if \(n \geq 4\), except for \(n = 4\) and except possibly \(n = 17, 18, 19, 21, 22\) and \(23\). Let \((3,2,1)\)-{ICOILS}\((v, k)\) denote an idempotent \((3,2,1)\)-COLS of order \(v\) with a hole of size \(k\). We provide \(15\) new \((3,2,1)\)-ICOILS\((v, k)\), where \(k = 2, 3,\) or \(5\).

A balanced part ternary design (BPTD) is a balanced ternary design (BTD) with a specified number of blocks, say \(b_2\), each having repeated elements. We prove some necessary conditions on \(b_2\) and show the existence of some particular BPTDs. We also give constructions of infinite families of BPTDs with \(b_1 = 0\), including families of ternary \(t\)-designs with \(t \geq 3\).