In 1992, Mahmoodian and Soltankhah conjectured that, for all \(0 \leq i \leq t\), a \((v, k, t)\) trade of volume \(2^{t+1} – 2^{t-i}\) exists. In this paper we prove this conjecture and, as a corollary, show that if \(s \geq (2t – 1)2^t\) then there exists a \((v, k, t)\) trade of volume \(s\).

We prove two new characterization theorems for finite Moufang polygons, one purely combinatorial, another group-theoretical. Both follow from a result of Andries Brouwer on the connectedness of the geometry opposite a flag in a finite generalized polygon.

Cyclonomial coefficients are defined as a generalization of binomial coefficients. It is proved that each natural number can be expressed, in a unique way, as the sum of cyclonomial coefficients, satisfying certain conditions. This cyclonomial number system generalizes the well-known binomial number system. It appears that this system is the appropriate number system to index the words of the lexicographically ordered code \(L^q(n, k)\). This code consists of all words of length \(n\) over an alphabet of \(q\) symbols, such that the sum of the digits is constant. It provides efficient algorithms for the conversion of such a codeword to its index, and vice versa.

We investigate the connections between families of graphs closed under (induced) subgraphs and their forbidden (induced) subgraph characterizations. In particular, we discuss going from a forbidden subgraph characterization of a family \(\mathbb{P}\) to a forbidden induced subgraph characterization of the family of line graphs of members of \(\mathbb{P}\) in the most general case. The inverse problem is considered too.

A family of double circulant quasi-cyclic codes is constructed from the incidence matrices of difference sets associated with hyperplanes in projective space. A subset of these codes leads to a class of doubly-even self-orthogonal codes, and two classes of self-dual codes.

All nonisomorphic \(2\)-\((21, 6, 3)\) designs with automorphisms of order \(7\) or \(5\) were found, and the orders of their groups of automorphisms were determined. There are \(33\) nonisomorphic \(2\)-\((21, 6, 3)\) designs with automorphisms of order \(7\) and \(203\) with automorphisms of order \(5\).

Let \(G\) be a graph with even order \(p\) and let \(k\) be a positive integer with \(p \geq 2k + 2\). It is proved that if the toughness of \(G\) is at least \(k\), then the subgraph of \(G\) obtained by deleting any \(2k – 1\) edges or \(k\) vertices has a perfect matching. Furthermore, we show that the results in this paper are best possible.

The following problem, known as the Strong Coloring Problem for the group \(G\) (SCP\(_G\)) is investigated for various permutation groups \(G\). Let \(G\) be a subgroup of \(S_h\), the symmetric group on \(\{0, \ldots, h-1\}\). An instance of SCP\(_G\) is an \(h\)-ary areflexive relation \(\rho\) whose group of symmetry is \(G\) and the question is “does \(\rho\) have a strong \(h\)-coloring”? Let \(m \geq 3\) and \(D_m\) be the Dihedral group of order \(m\). We show that SCP\(_{D_m}\) is polynomial for \(m = 4\), and NP-complete otherwise. We also show that the Strong Coloring Problem for the wreath product of \(H\) and \(K\) is in \( {P}\) whenever both SCP\(_H\) and SCP\(_K\) are in \( {P}\). This, together with the algorithm for \(D_4\) yields an infinite new class of polynomially solvable cases of SCP\(_G\).

We deal with the concept of packings in graphs, which may be regarded as a generalization of the theory of graph design. In particular, we construct a vertex- and edge-disjoint packing of \(K_n\) (where \(\frac{n}{2} \mod 4\) equals 0 or 1) with edges of different cyclic length. Moreover, we consider edge-disjoint packings in complete graphs with uniform linear forests (and the resulting packings have special additional properties). Further, we give a relationship between finite geometries and certain packings which suggests interesting questions.

A homomorphism from a graph to another graph is an edge preserving vertex mapping. A homomorphism naturally induces an edge mapping of the two graphs. If, for each edge in the image graph, its preimages have \(k\) elements, then we have an edge \(k\)-to-\(1\) homomorphism. We characterize the connected graphs which admit edge \(2\)-to-\(1\) homomorphism to a path, or to a cycle. A special case of edge \(k\)-to-\(1\) homomorphism — \(k\)-wrapped quasicovering — is also considered.