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.

We enumerate the 2-\((9,4,6)\) designs and find 270,474,142 non-isomorphic such designs in a backtrack search. The sizes of their automorphism groups vary between 1 and 360. Out of these designs, 19,489,464 are simple and 2,148,676 are decomposable.

A \(t\)-partite number is a \(t\)-tuple \(\vec{n} = (n_1, \ldots, n_t)\), where \(n_1, \ldots, n_t\) are positive integers. For a \(t\)-partite number \(\vec{n}\), let \(f_t(\vec{n})\) be the number of different ways to write \(\vec{n}\) as a product of \(t\)-partite numbers, where the multiplication is performed coordinate-wise, \((1, 1, \ldots, 1)\) is not used as a factor of \(\vec{n}\), and two factorizations are considered the same if they differ only in the order of the factors. This paper gives the following explicit upper bound for the multiplicative partition function \(f_t(\vec{n})\):
\[f_t(n_1, \ldots, n_t) \leq M^{w(t)},\, \text{where}\,\, M = \Pi_{i=1}^t n_i \,\,\text{and}\,\, w(t) = \frac{\log((t+1)1)}{t\log2}\].

The following partition problem was first introduced by R.C. Entringer and has subsequently been studied by the first author and more recently by Bollobas and Scott, who consider the hypergraph version as well, using a probabilistic technique. The partition problem is that of coloring the vertex set of a graph with \(s\) colors so that the number of induced edges is bounded for each color class. The techniques employed are non-constructive and non-probabilistic and improve the known bounds in the previous papers.