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.