Given the number of vertices \( n \), labelled graphs can easily be generated uniformly at random by simply selecting each edge independently with probability \( \frac{1}{2} \). With \( \frac{n(n-1)}{2} \) processors, this takes constant parallel time. In contrast, the problem of uniformly generating unlabelled graphs of size \( n \) is not so straightforward. In this paper, we describe an efficient parallelisation of a classic algorithm of Dixon and Wilf for the uniform generation of unlabelled graphs on \( n \) vertices. The algorithm runs in \( O(\log n) \) expected time on a CREW PRAM using \( n^2 \) processors.

We discuss a parallel programming method for solving the maximum clique problem. We use the partitioned shared memory programming language, Unified Parallel C, for the parallel implementation. The problem of load balancing is discussed and the steal stack is used to solve this problem. Implementation details are provided.

A \( 4 \)-cycle system of order \( n \) is said to be almost resolvable provided its \( 4 \)-cycles can be partitioned into \( \frac{n-1}{2} \) almost parallel classes (i.e., \( \frac{n-1}{4} \) vertex-disjoint \( 4 \)-cycles) and a half parallel class (i.e., \( \frac{n-1}{8} \) vertex-disjoint \( 4 \)-cycles). We construct an almost resolvable \( 4 \)-cycle system of every order \( n \equiv 1 \pmod{8} \) except \( 9 \) (for which no such system exists) and possibly \( 33, 41, \) and \( 57 \).

Splitting balanced incomplete block designs were first formulated by Ogata, Kurosawa, Stinson, and Saido recently in the investigation of authentication codes. This article investigates the existence of splitting balanced incomplete block designs, i.e., \( (v, 2k, \lambda) \)-splitting BIBDs; we give the spectrum of \( (v, 2 \times 4, \lambda) \)-splitting BIBDs.

In this paper, we first present new proofs, much shorter and much simpler than can be found elsewhere, of two facts about Hypercubes: that for the \( d \)-dimensional Hypercube, there exist sets of paths by which any \emph{permutation routing} task may be accomplished in at most \( 2d – 1 \) steps without queueing; and, when \( d \) is even, there exists an edge decomposition of the Hypercube into precisely \( \frac{d}{2} \) edge-disjoint Hamiltonian cycles. The permutation routing paths are computed off-line. Whether or not these paths may be computed by an online parallel algorithm in \( O(d) \)-time has long been an open question. We conclude by speculating on whether the use of a Hamiltonian decomposition of the Hypercube might lead to such an algorithm.

The search for special substructures in combinatorial objects that have a lot of symmetry, such as searching for maximal partial ovoids or spreads in generalized quadrangles, can often be translated to a well-known algorithmic problem, such as a maximum clique problem in a graph. These problems are typically NP-hard. However, using standard backtracking strategies together with pruning techniques based on problem-specific properties, it is possible to obtain non-trivial results which are mathematically interesting. In some cases, heuristic techniques can also lead to interesting results. In this paper, we describe some techniques as well as new results obtained for maximal partial ovoids and spreads in generalized quadrangles.

Built on earlier works of Larcombe on a certain class of non-terminating expansions of the sine function, we set up two new \( {_{}{3}F_2} \) summation formulas via integration.

In this paper, we investigate exhaustively the cyclically indecomposable triple systems \( TS_\lambda(v) \) for \( \lambda = 2, v \leq 33 \) and \( \lambda = 3, v \leq 21 \), and we identify the decomposable ones. We also construct, by using Skolem-type and Rosa-type sequences, cyclically indecomposable two-fold triple systems \( TS_2(v) \) for all admissible orders. Further, we investigate exhaustively all cyclic \( TS_2(v) \) that are constructed by Skolem-type and Rosa-type sequences up to \( v \leq 45 \) for indecomposability.

We show that if the independence number of a graph is \( \alpha \), then the eternal security number of the graph is at most \( \binom{\alpha+1}{2} \), solving a problem stated by Goddard, Hedetniemi, and Hedetniemi \([JCMCC, \text{ vol. } 52, \text{ pp. } 160-180]\).