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A Formulation of Quantum Stochastic Processes and Some of its Properties. Hellwig, K.-E.; Stulpe, W.
(1983)
An equitable graph coloring is a proper vertex coloring of a graph G where the sizes of the color classes differ by at most one. The equitable chromatic number is the smallest number k such that G admits such equitable k-coloring. We focus on enumerative algorithms for the computation of the equitable coloring number and propose a general scheme to derive pruning rules for them: We show how the extendability of a partial coloring into an equitable coloring can be modeled via network flows. Thus, we obtain pruning rules which can be checked via flow algorithms. Computational experiments show that the search tree of enumerative algorithms can be significantly reduced in size by these rules and, in most instances, such naive approach even yields a faster algorithm. Moreover, the stability, i.e., the number of solved instances within a given time limit, is greatly improved.
Since the execution of flow algorithms at each node of a search tree is time consuming, we derive arithmetic pruning rules (generalized Hall-conditions) from the network model. Adding these rules to an enumerative algorithm yields an even larger runtime improvement.
Manufacturing process simulation enables the evaluation and improvement of autoclave mold concepts early in the design phase. To achieve a high part quality at low cycle times, the thermal behavior of the autoclave mold can be investigated by means of simulations. Most challenging for such a simulation is the generation of necessary boundary conditions. Heat-up and temperature distribution in an autoclave mold are governed by flow phenomena, tooling material and shape, position within the autoclave, and the chosen autoclave cycle. This paper identifies and summarizes the most important factors influencing mold heat-up and how they can be introduced into a thermal simulation. Thermal measurements are used to quantify the impact of the various parameters. Finally, the gained knowledge is applied to develop a semi-empirical approach for boundary condition estimation that enables a simple and fast thermal simulation of the autoclave curing process with reasonably high accuracy for tooling optimization.