In this paper, finite \( \{2,t\} \)-semiaffine linear spaces are investigated. When \( t = 5 \), their parameters are determined, and it is also proved that there is a single finite \( \{2, 5\} \)-semiaffine linear space on \( v = 20 \) points and with constant point degree \( 7 \).

We establish that for each of the 5005 possible types of 2-factorizations of the complete graph \( K_{13} \), there exists at least one solution. We also enumerate all nonisomorphic solutions to the Oberwolfach problem \( \text{OP}(13;3,3,3,4) \).

Scheduling static tasks on parallel architectures is a basic problem arising in the design of parallel algorithms. This NP-complete problem has been widely investigated in the literature and remains one of the most challenging questions in the field. Among the resolution methods for this type of problems, the taboo search technique is of particular interest. Based on this technique, two algorithms are proposed and tested on a sample of instances in order to be compared experimentally with other well-known algorithms. The results clearly indicate good overall performances of our algorithms. Next, some NP-completeness results are established showing that this problem is intractable for approximation, even for some restricted cases bearing a clear relation to the instances treated experimentally in this work.

\( X \)-proper edge colourings of bipartite graphs are defined. These colourings arise in timetables where rooms have to be assigned to courses. The objective is to minimize the number of different rooms in which each course must be taught. An optimum assignment is represented by a \( k \)-optimum edge colouring of a bipartite graph. Some necessary conditions for a \( k \)-optimum colouring are obtained, in terms of forbidden subgraphs. An algorithm based on removing these forbidden subgraphs to obtain improved colourings is described.

A graph \( G \) of order \( n \) is pancyclic if it contains a cycle of length \( \ell \) for every \( \ell \) such that \( 3 \leq \ell \leq n \). If the graph is bipartite, then it contains no cycles of odd length. A balanced bipartite graph \( G \) of order \( 2n \) is bipancyclic if it contains a cycle of length \( \ell \) for every even \( \ell \), such that \( 4 \leq \ell \leq 2n \). A graph \( G \) of order \( n \) is called \( k \)-semipancyclic, \( k \geq 0 \), if there is no “gap” of \( k+1 \) among the cycle lengths in \( G \), i.e., for no \( \ell \leq n-k \) is it the case that each of \( C_\ell, \ldots, C_{\ell+k} \) is missing from \( G \). Generalizing this to bipartite graphs, a bipartite graph \( G \) of order \( n \) is called \( k \)-semibipancyclic, \( k \geq 0 \), if there is no “gap” of \( k+1 \) among the even cycle lengths in \( G \), i.e., for no \( \ell \leq n-2k \) is it the case that each of \( C_{2\ell}, \ldots, C_{2\ell+2k} \) is missing from \( G \).

In this paper we generalize a result of Hakimi and Schmeichel in several ways. First to \( k \)-semipancyclic, then to bipartite graphs, giving a condition for a hamiltonian bipartite graph to be bipancyclic or one of two exceptional graphs. Finally, we give a condition for a hamiltonian bipartite graph to be \( k \)-semibipancyclic or a member of a very special class of hamiltonian bipartite graphs.

We consider labeling edges of graphs with elements from abelian groups. Particular attention is given to graphs where the labels on any two Hamiltonian cycles sum to the same value. We find several characterizations for such labelings for cubes, complete graphs, and complete bipartite graphs. This extends work of \([1, 8, 9, 10]\). We also consider the computational complexity of testing if a labeled graph has this property and show it is NP-complete even when restricted to integer labelings of 3-connected, cubic, planar graphs with face girth at least five.

We investigate the optimization of a real-world logistics problem, which is concerned with shipping a dangerous chemical substance in various degrees of refinement to several locations and customers. Transport frequencies, inventories, and container flows have to be optimized. On the one hand, we discuss the mathematical structure of our problem (one result being its NP-completeness), and on the other hand, we describe our practical approach, which achieves nearly optimal solutions.

Let \( G \) be a \( k \)-regular graph of odd order \( n \geq 3 \) with \( k \geq \frac{n + 1}{2} \). This implies that \( k \) is even. Furthermore, let
\[
p = \min\left\{\frac{k}{2}, \left\lceil k-\frac{n}{3}\right\rceil\right\}.
\]
If \( x_1, x_2, \ldots, x_p \) are arbitrary given, pairwise different, vertices of the graph \( G \), then we show in this paper that there exist \( p \) pairwise edge-disjoint almost perfect matchings \( M_1, M_2, \ldots, M_p \) in \( G \) with the property that no edge of \( M_i \) is incident with \( x_i \) for \( i = 1, 2, \ldots, p \).

The previously studied notions of smart and foolproof finite order domination of a simple graph \( G = (V, E) \) are generalised in the sense that safe configurations in \( G \) are not merely sought after \( k \geq 1 \) moves, but in the limiting cases where \( k \to \infty \). Some general properties of these generalised domination parameters are established, after which the parameter values are found for certain simple graph structures (such as paths, cycles, multipartite graphs, and products of complete graphs, cycles, and paths).

Self-dual codes are an important class of linear codes. Hadamard matrices and weighing matrices have been used widely in the construction of binary and ternary self-dual codes. Recently, weighing matrices and orthogonal designs have been used to construct self-dual codes over larger fields. In this paper, we further investigate codes over \( \mathbb{F}_p \), constructed from orthogonal designs. Necessary conditions for these codes to be self-dual are established, and examples are given for lengths up to 40. Self-dual codes of lengths \( 2n \geq 16 \) over \( GF(31) \) and \( GF(37) \) are investigated here for the first time. We also show that codes obtained from orthogonal designs can generally give better results, with respect to their minimum Hamming distance, than codes obtained from Hadamard matrices, weighing matrices, or conference matrices.