It is shown that the necessary conditions are sufficient for the existence of all \(c\)-BRDs\((v, 3, \lambda)\) for negative \(c\)-values. This completes the study of \(c\)-BRDs with block size three as previously the authors and J. Seberry have shown that the necessary conditions are sufficient for \(c \geq -1\).

Let \(G\) be a simple connected graph on \(2n\) vertices with a perfect matching. For a positive integer \(k\), \(1 \leq k \leq n-1\), \(G\) is \(k\)-\emph{extendable} if for every matching \(M\) of size \(k\) in \(G\), there is a perfect matching in \(G\) containing all the edges of \(M\). For an integer \(k\), \(0 \leq k \leq n – 2\), \(G\) is \emph{strongly \(k\)-extendable} if \(G – \{u, v\}\) is \(k\)-extendable for every pair of vertices \(u\) and \(v\) of \(G\). The problem that arises is that of characterizing \(k\)-extendable graphs and strongly \(k\)-extendable graphs. The first of these problems has been considered by several authors whilst the latter has been investigated only for the case \(k = 0\). In this paper, we focus on the problem of characterizing strongly \(k\)-extendable graphs for any \(k\). We present a number of properties of strongly \(k\)-extendable graphs including some necessary and sufficient conditions for strongly \(k\)-extendable graphs.

In this paper we count the number of non-homeomorphic continua in a certain collection of continua. The continua in these collections are trees with certain restrictions on them. We refer to a continuum in one of these collections as a caterpillar continuum.

A connected graph \(G = (V, E)\) is \((a, d)\)-antimagic if there exist positive integers \(a, d\) and a bijection \(g: E \to \{1, 2, \ldots, |E|\}\) such that the induced mapping
\[f_g = \Sigma\{g(u,v): (u, v) \in E(G)\}\, \text{is injective and}\]
\[f_g(V) = \{a, a+d, a+2d, \ldots, a+(|V|-1)d\}.\]
In this paper, we prove two conjectures of Baca concerning \((a, d)\)-antimagic labelings of antiprisms

Some special sum graphs and difference graphs, based on abelian groups, are discussed. In addition to Li’s result on character sum estimates, Weil’s character sum estimates are also used to show that these are indeed Ramanujan graphs.

A critical set in a Latin square of order \(n\) is a set of entries in a Latin square which can be embedded in precisely one Latin square of order \(n\). Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order \(n\). A smallest critical set in a Latin square is a critical set of minimum cardinality. In this paper we find smallest critical sets for all the Latin squares of orders six and seven. We also find smallest critical sets of orders six and seven which are also weak critical sets. In particular, we find a weak critical set of size twelve for the dihedral group of order six.

We study combinatorial structure of \(\ell\)-optimal \(A^2\)-codes that offer the best protection for spoofing of order up to \(\ell\) and require the least number of keys for the transmitter and the receiver. We prove that for such codes the transmitter’s encoding matrix is a strong partially balanced resolvable design, and the receiver’s verification matrix corresponds to an \(\alpha\)-resolvable design with special properties.

It is proved in this paper that for any integer \(n \geq 136\), a SODLS(\(v, n\)) (self-orthogonal diagonal Latin square with missing subsquare) exists if and only if \(v \geq 3n+2\) and \(v-n\) even.

Employing trading signed design algorithm, we construct an automorphism-free \(4\)-\((15, 5, 5)\) design.

Consider those graphs \(G\) of size \(2n\) that have an eigenvalue \(\lambda\) of multiplicity \(n\) and where the edges between the star set and its complement is a matching. We show that \(\lambda\) must be either \(0\) or \(1\) and completely characterize the corresponding graphs.