Chvátal conjectured that if \(G\) is a \(k\)-tough graph and \(k|V(G)|\) is even, then \(G\) has a \(k\)-factor. In \([5\) it was proved that Chvátal’s conjecture is true. Katerinis\([2]\) presented a toughness condition for a graph to have an \([a, b]\)-factor. In this paper, we prove a stronger result: every \((a – 1 + a/b)\)-tough graph satisfying all necessary conditions has an \([a, b]\)-factor containing any given edge and another \([a, b]\)-factor excluding it. We also discuss some special cases of the above result.

R.A. Bailey has conjectured that all finite groups except elementary Abelian \(2\)-groups with more than one factor have \(2\)-sequencings (i.e., terraces). She verified this for all groups of order \(n\), \(n \leq 9\). Results proved since the appearance of Bailey’s paper make it possible to raise this bound to \(n \leq 87\) with \(n = 64\) omitted. Relatively few groups of order not \(2^n\), \(n \in \{4, 5\}\) must be handled by machine computation.

A set \(S\) of vertices of a graph \(G = (V, E)\) is a global dominating set if \(S\) dominates both \(G\) and its complement \(\overline{G}\). The concept of global domination was first introduced by Sampathkumar. In this paper, we extend this notion to irredundancy. A set \(S\) of vertices will be called universal irredundant if \(S\) is irredundant in both \(G\) and \(\overline{G}\). A set \(S\) will be called global irredundant if for every \(x\) in \(S\), \(x\) is an irredundant vertex in \(S\) either in \(G\) or in \(\overline{G}\). We investigate the universal irredundance and global irredundance parameters of a graph. It is also shown that the determination of the upper universal irredundance number of graphs is NP-Complete.

We enumerate by computer algorithms all simple \(t-(t+7, t+1, 2)\) designs for \(1 \leq t \leq 5\), i.e., for all possible \(t\). This enumeration is new for \(t \geq 3\). The number of nonisomorphic designs is equal to \(3, 13, 27, 1\) and \(1\) for \(t = 1, 2, 3, 4\) and \(5\), respectively. We also present some properties of these designs, including orders of their full automorphism groups and resolvability.

Let \(G\) be a finite simple graph. The vertex clique covering number \({vcc}(G)\) of \(G\) is the smallest number of cliques (complete subgraphs) needed to cover the vertex set of \(G\). In this paper, we study the function \({vcc}(G)\) for the case when \(G\) is \(r\)-regular and \((r-2)\)-edge-connected. A sharp upper bound for \({vcc}(G)\) is determined. Further, the set of possible values of \({vcc}(G)\) when \(G\) is a \(4\)-regular connected graph is determined.

We consider certain resolvable designs which have applications to doubly perfect Cartesian authentication schemes. These generalize structures determined by sets of mutually orthogonal Latin squares and are related to semi-Latin squares and other designs which find applications in the design of experiments.

A \(1\)-spread of a BIBD \(\mathcal{D}\) is a set of lines of maximal size of \(\mathcal{D}\) which partitions the point set of \(\mathcal{D}\). The existence of infinitely many non-symmetric BIBDs which (i) possess a \(1\)-spread, and (ii) are not merely a multiple of a symmetric BIBD,
is shown. It is also shown that a \(1\)-spread \(\mathcal{S}\) gives rise to a regular group divisible design \(\mathcal{G}(\mathcal{S})\). Necessary and sufficient conditions that the dual of such a group divisible design \(\mathcal{G}(\mathcal{S})\) be a group divisible design are established and used to show the existence of an infinite class of symmetric regular group divisible designs whose duals are not group divisible.

We consider the changing and unchanging of the edge covering and edge independence numbers of a graph when the graph is modified by deleting a node, deleting an edge, or adding an edge. In this paper, we present characterizations for the graphs in each of these classes and some relationships among them.

Let \(G\) be the automorphism group of an \((3, 5, 26)\) design. We show the following: (i) If \(13\) divides \(|G|\), then \(G\) is a subgroup of \(Z_2 \times F_{r_{13 \cdot 12}}\), where \(F_{r_{13 \cdot 12}}\) is the Frobenius group of order \(13 \cdot 12\); (ii)If \(5\) divides \(|G|\), then \(G \cong {Z}_5\) or \(G \cong {D}_{10}\); and (iii) Otherwise, either \(|G|\) divides \(3 \cdot 2^3\) or \(2^4\).

We investigate the edge-gracefulness of \(2\)-regular graphs.

For \(n\) a positive integer and \(v\) a vertex of a graph \(G\), the \(n\)th order degree of \(v\) in \(G\), denoted by \(\text{deg}_n(v)\), is the number of vertices at distance \(n\) from \(v\). The graph \(G\) is said to be \(n\)th order regular of degree \(k\) if, for every vertex \(v\) of \(G\), \(\text{deg}_n(v) = k\). For \(n \in \{7, 8, \ldots, 11\}\), a characterization of \(n\)th order regular trees of degree \(2\) is obtained. It is shown that, for \(n \geq 2\) and \(k \in \{3, 4, 5\}\), if \(G\) is an \(n\)th order regular tree of degree \(k\), then \(G\) has diameter \(2n – 1\).

We prove that there exist precisely \(459\) pairwise non-isomorphic Steiner systems \(S(5,6,48)\) stabilized by the group \({PSL}_2(47)\).

The known generalized quadrangles with parameters \((s,t)\) where \(|s-t| = 2\) have been characterized in several ways by M. De Soete \([D]\), M. De Soete and J. A. Thas \([DT1]\), \([DT2]\), \([DT4]\), and the present author \([P]\). Certain of these results are interpreted for a coset geometry construction.

In this paper, we illustrate the relationship between profiles of Hadamard matrices and weight distributions of codes, give a new and efficient method to determine the minimum weight \(d\) of doubly even self-dual \([2n,n,d]\) codes constructed by using Hadamard matrices of order \(n = 8t + 4\) with \(t \geq 1\), and present a new proof that the \([2n,n,d]\) codes have \(d \geq 8\) for all types of Hadamard matrices of order \(n = 8t + 4\) with \(t \geq 1\). Finally, we discuss doubly even self-dual \([72,36,d]\) codes with \(d = 8\) or \(d = 12\) constructed by using all currently known Hadamard matrices of order \(n = 36\).

We define an \({extremal \; graph}\) on \(v\) vertices to be a graph that has the maximum number of edges on \(v\) vertices, and that contains neither \(3\)-cycles nor \(4\)-cycles.
We establish that every vertex of degree at least \(3\), in an extremal graph of at least \(7\) vertices, is in a \(5\)-cycle; we enumerate all of the extremal graphs on \(21\) or fewer vertices; and we determine the size of extremal graphs of orders \(25\), \(26\), and \(27\).

We consider square arrays of numbers \(\{a(n, k)\}\), generalizing the binomial coefficients:
\(a(n, 0) = c_n\), where the \(c_n\) are non-negative real numbers; \(a(0, k) = c_0\), and if \(n, k > 0\), then \(a(n, k) = a(n, k – 1) + a(n – 1, k)\).
We give generating functions and arithmetical relations for these numbers. We show that every row of such an array is eventually log concave, and give a few sufficient conditions for columns to be eventually log concave. We also give a necessary condition for a column to be eventually log concave, and provide examples to show that there exist such arrays in which no column is eventually log concave.

In this paper, we obtain some necessary conditions for the existence of balanced arrays (\(B\)-arrays) of strength \(4\) and with two levels, and we state the usefulness of these conditions in obtaining an upper bound on the number of constraints for these B-arrays.

It is shown that the circuit polynomial of a graph, when weighted by the number of nodes in the circuits, does not characterize the graph, i.e., non-isomorphic graphs can have the same circuit polynomial. Some general theorems are given for constructing graphs with the same circuit polynomial (cocircuit graphs). Analogous results can be deduced for characteristic polynomials.

Given \(n\) real numbers whose sum is zero, find one of the numbers that is non-negative. In the model under consideration, an algorithm is allowed to compute \(p\) linear forms in each time step until it knows an answer. We prove that exactly \(\lceil{\log n}/{\log(p+1)} \rceil\) time steps are required. Some connections with parallel group-testing problems are pointed out.