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.

In a communication network, several vulnerability measures are used to determine the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modeling a network, the edge-integrity of a graph is one \(\textbf{measure of graph vulnerability}\) and it is defined to be the minimum sum of the orders of a set of edges being removed and a largest remaining component. In this paper, the edge-integrity of graphs \(B_n\), \(H_n\), and \(E_p^t\), are calculated. Also, some results are given about edge-integrity of these graphs.

In this paper, it is shown that the necessary condition for the existence of a holey perfect Mendelsohn design (HPMD) with block size 5, type \(h^n\) and index \(\lambda\), namely, \(n \geq 5\) and \(\lambda n(n-1)h^2 \equiv 0 \pmod{5}\), is also sufficient for \(\lambda \geq 2\). The result guarantees the analogous existence result for group divisible designs (GDDs) of type \(h^n\) having block size 5 and index \(4\lambda\).

The computational complexity of the graph isomorphism problem is still unknown. We consider Cartesian products \(K_n \times K_m\) of two complete graphs \(K_n\) and \(K_m\). An acyclic orientation of such a Cartesian product is called a sequence graph because it has an application in production scheduling. It can be shown that the graph isomorphism problem on the class of these acyclic digraphs is solvable in polynomial time. We give numbers of non-isomorphic sequence graphs for small \(n\) and \(m\). The orientation on the cliques of a sequence graph can be interpreted as job orders and machine orders of a shop scheduling problem with a complete operation set.