We study the problem of scheduling parallel programs with conditional branching on parallel processors. The major problem in having conditional branching is the non-determinism since the direction of a branch may be unknown until the program is midway in execution. In this paper, we overcome the problem of non-determinism by proposing a probabilistic model that distinguishes between branch and precedence relations in parallel programs. We approach the problem of scheduling task graphs that contain branches by representing all possible execution instances of the program by a single deterministic task graph, called the representative task graph. The representative task graph can be scheduled using one of the scheduling techniques used for deterministic cases. We also show that a schedule for the representative task graph can be used to obtain schedules for all execution instances of the program.

We give a list of all graphs of maximum degree three and order at most sixteen which are critical with respect to the total chromatic number.

In this paper we give a partial answer to a query of Lindner conceming the quasigroups arising from \(2\)-perfect \(6\)-cycle systems.

Consider the paths \(\pi_t(i_1), \ldots, \pi_t(i_k)\) from the root to the leaves \(i_1, \ldots, i_k\) in a random binary tree \(t\) with \(n\) internal nodes, where all such trees are assumed equally likely and the leaves are enumerated from left to right. We investigate, for fixed \(i_1, \ldots, i_k\) and \(n\), the average size of \(\pi_t(i_1)\cup \ldots \cup \pi_t(i_k)\) resp. of \(\pi_t(i_1)\cap \ldots \cap \pi_t(i_k)\) (the latter corresponding to the average depth of the smallest subtree containing \(i_1, \ldots, i_k\)). By a rotation argument, both problems are reduced to the case \(k=1\), for which a solution is known. Furthermore, formulas for the probability distributions of the depth of leaf \(i\), the distance between leaf \(i\) and \(j\) and the length of \(\pi_t(i) \cap \pi_t(j)\) are derived.

Chetwynd and Hilton made the following \({edge-colouring \; conjecture}\): if a simple graph \(G\) satisfies \(\Delta(G) > \frac{1}{3}|V(G)|\), then \(G\) is Class \(2\) if and only if it contains an overfull subgraph \(H\) with \(\Delta(H) = \Delta(G)\). They also made the following \({total-colouring \; conjecture}\): if a simple graph \(G\) satisfies \(\Delta(G) \geq \frac{1}{2}(|V(G)|+ 1)\), then \(G\) is Type \(2\) if and only if \(G\) contains a non-conformable subgraph \(H\) with \(\Delta(H) = \Delta(G)\). Here we show that if the edge-colouring conjecture is true for graphs of even order satisfying \(\Delta(G) > \frac{1}{2}|V(G)|\), then the total-colouring conjecture is true for graphs of odd order satisfying \(\delta(G) \geq \frac{3}{4}{|V(G)|} – \frac{1}{4}\) and \(\text{def}(G) \geq 2(\Delta(G) – \delta(G) + 1)\).

We correct an omission by Mathon in his classification of symmetric \((31, 10, 3)\)-designs with a non-trivial automorphism group and find that there are a further six such designs, all with an automorphism group of order \(3\).

A \({dominating \; function}\) is a feasible solution to the LP relaxation of the minimum dominating set \(0-1\) integer program. A minimal dominating function (MDF) g is called universal if every convex combination of g and any other MDF is also a MDF. The problem of finding a universal MDF in a tree \({T}\) can also be described by a linear program. This paper describes a linear time algorithm that finds a universal MDF in \({T}\), if one exists.

Let \(H\) be a digraph whose vertices are called colours. Informally, an \(H\)-colouring of a digraph \(G\) is an assignment of these colours to the vertices of \(G\) so that adjacent vertices receive adjacent colours. In this paper we continue the study of the \(H\)-colouring problem, that is, the decision problem “Does there exist an \(H\)-colouring of a given digraph \(G\)?”. In particular, we prove that the \(H\)-colouring problem is NP-complete if the digraph \(H\) consists of a directed cycle with two chords, or two directed cycles joined by an oriented path, or is obtained from a directed cycle by replacing some arcs by directed two-cycles, so long as \(H\) does not retract to a directed cycle. We also describe a new reduction which yields infinitely many new infinite families of NP-complete \(H\)-colouring problems.