We study the maximal intersection number of known Steiner systems and designs obtained from codes. By using a theorem of Driessen, together with some new observations, we obtain many new designs.

Taking as blocks some subspace pairs in a finite unitary geometry, we construct a number of new Balanced Incomplete Block (BIB) designs and Partially Balanced Incomplete Block (PBIB) designs, and also give their parameters.

The support size of a factorization is the sum over the factors of the number of distinct edges in each factor. The spectrum of support sizes of \(l\lambda\)-factorizations of \(\lambda K_n\) and \(\lambda K_{n,n}\) are completely determined for all \(\lambda\) and \(n\).

Latin interchanges have been used to establish the existence of critical sets in Latin squares, to search for subsquare-free Latin squares, and to investigate the intersection sizes of Latin squares. Donald Keedwell was the first to study Latin interchanges in their own right. This paper builds on Keedwell’s work and proves new results about the existence of Latin interchanges.

A balanced ternary design of order nine with block size three, index two, and \(\rho_2 = 1\) is a collection of multi-subsets of size \(3\) (of type \(\{x, y, z\}\) or \(\{x, x, y\}\)) called blocks, chosen from a \(9\)-set, in which each unordered pair of distinct elements occurs twice, possibly in one block, and in which each element is repeated in just one block. So there are precisely \(9\) blocks of type \(\{x, x, y\}\). We denote such a design by \((9; 1; 3, 2)\) BTD. In this note, we describe the procedures we have used to
determine that there are exactly \(1475\) non-isomorphic \((9; 1; 3, 2)\) BTDs.

A graph \(G\) is said to be \({hypohamiltonian}\) if \(G\) is not Hamiltonian but for each \(v \in V(G)\), the vertex-deleted subgraph \(G – v\) is Hamiltonian. In this paper, we show that there is no hypohamiltonian graph on \(17\) vertices and thereby complete the answer to the question, “For which values of \(n\) do there exist hypohamiltonian graphs on \(n\) vertices?”. In addition, we present an exhaustive list of hypohamiltonian graphs on fewer than \(18\) vertices and extend previously obtained results for cubic hypohamiltonian graphs.

We consider the problem of constructing pairwise balanced designs of order \(v\) with a hole of size \(k\). This problem was addressed by Hartman and Heinrich who gave an almost complete solution. To date, there remain fifteen unresolved cases. In this paper, we construct designs settling all of these.

All non-Hamiltonian cubic \(2\)-edge-connected graphs, including all snarks, on \(16\) or fewer vertices are listed, along with some of their properties. Questions concerning the existence of graphs with certain properties are posed.

We deal with finite graphs which admit a labeling of edges by pairwise different positive integers from the set \(\{1, 2, \ldots, |E(G)|\}\) in such a way that the sum of the labels of the edges incident to a particular vertex is the same for all vertices. We construct edge labelings for two families of quartic graphs, i.e., regular graphs of degree \(d = 4\).

Kibler, Baumert, Lander, and Kopilovich (cf. [7], [1], [10], and [8] respectively), studied the existence of \( (v, k, \lambda) \)-abelian difference sets with \( k \leq 100 \). In Lander and Kopilovich’s works, there were \( 13 \) and \( 8 \) \( (v, k, \lambda) \) tuples, respectively, in which the problem was open. Later, several authors have completed these studies and nowadays the problem is open for \( 6 \) and \( 7 \) tuples, respectively. Jungnickel (cf. [9]) lists some unsolved problems on difference sets. One of them is to extend Lander’s table somewhat. By following this idea, this paper deals with the existence or non-existence of \( (v, k, \lambda) \)-abelian difference sets with \( 100 < k \leq 150 \). There exist \( 277 \) tuples that satisfy the basic relationship between the parameters \( v \), \( k \), and \( \lambda \), \( k \leq v/2 \), Schutzenberger and Bruck-Chowla-Ryser's necessary conditions, and \( 100 < k \leq 150 \). In order to reduce this number, we have written in C several programs which implement some known criteria on non-existence of difference sets. We conclude that the only \( (v, k, \lambda) \) tuples, with \( 100 < k \leq 150 \), for which a difference set in some abelian group of order \( v \) can exist are \begin{align*} &(10303, 102, 1), (10713, 104, 1), (211, 105, 52), (11557, 108, 1), \\ &(223, 111, 55), (11991, 110, 1), (227, 113, 56), (12883, 114, 1), \\ &(378, 117, 386), (239, 119, 59), (256, 120, 56), (364, 121, 40), \\ &(243, 121, 60), (14763, 122, 1), (251, 125, 62), (15751, 126, 1), \\ &(351, 126, 45), (255, 127, 63), (16257, 128, 1), (16513, 129, 1), \\ &(263, 131, 65), (17293, 132, 1), (1573, 132, 11), (1464, 133, 12), \\ &(271, 135, 67), (18907, 138, 1), (19461, 140, 1), (283, 141, 70), \\ &(22351, 150, 1), (261, 105, 42), (429, 198, 27), (1200, 110, 10), \\ &(768, 118, 18), (841, 120, 17), (715, 120, 20), (5085, 124, 3), \\ &(837, 133, 21), (419, 133, 42), (1225, 136, 15), (361, 136, 51), \\ &(1975, 141, 10), (1161, 145, 18), (465, 145, 45), (5440, 148, 4), \\ &(448, 150, 50). \end{align*} It is known that there exist difference sets for the first \( 29 \) tuples and the problem is open for the remaining \( 16 \). Besides, in Table 1, we give the criterion that we have applied to determine the non-existence of \( (v, k, \lambda) \)-difference sets for the rest of the tuples.