A holey perfect Mendelsohn design of type \(h_1^{n_1} h_2^{n_2} \ldots h_k^{n_k}\) (HPMD\((h_1^{n_1} h_2^{n_2} \ldots h_k^{n_k})\)), with block size four is equivalent to a frame idempotent quasigroup satisfying Stein’s third law with the same type, where a frame quasigroup of type \(h_1^{n_1} h_2^{n_2} \ldots h_k^{n_k}\) means a quasigroup of order \(n\) with \(n_i\) missing subquasigroups (holes) of order \(h_i\), \(1 \leq i \leq k\), which are disjoint and spanning, that is \(\sum_{1\leq i \leq k} n_ih_i = n\). In this paper, we investigate the existence of HPMD\((2^n3^1)\) and show that an HPMD\((2^n3^1)\) exists if and only if \(n \geq 4\). As an application, we readily obtain HSOLS\((2^n3^1)\) and establish the existence of \((2,3,1)\) [or \((3,1,2)\)]-HCOLS\((2^n3^1)\) for all \(n \geq 4\).

Much of chordal graph theory and its applications is based on chordal graphs being the intersection graphs of subtrees of trees. This suggests also looking at intersection graphs of subgraphs of chordal graphs, and so on, with appropriate conditions imposed on the subgraphs.This paper investigates such a hierarchy of generalizations of “chordal-type” graphs, emphasizing the so-called “ekachordal graphs” — those next in line beyond chordal graphs. Parts of the theory of chordal graphs do carry over to chordal-type graphs, including a recursive, elimination characterization for ekachordal graphs.

We present a new construction to obtain frames with block size four using certain skew Room frames. The existence results of Rees and Stinson for frames with block size four are improved, especially nfor hole sizes divisible by \(6\). As a by-product of the skew Room frames we construct, we are also able to show that a resolvable \((K_4 – e)\)-design with \(60t + 16\) points exists if \(t \geq 0\) and \(t \neq 8, 12\).

It has been proved that the smallest rectangular board on which a \( (p, q) \)-knight’s graph is connected has sides \( p+q \) by \( 2q \) when \( p < q \). It has also been proved that these minimal connected knight's graphs are of genus \( 0 \) or \( 1 \), and that they are of genus \( 0 \) when \( p \) and \( q \) are of the form \( Md+1 \) and \( (M+1)d+1 \), with \( M \) a non-negative integer and \( d \) a positive odd integer. It is proved in this paper that the minimal connected knight's graph is of genus \( 1 \) in all other cases.

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.