Lovász had posed a question stating whether every connected, vertex-transitive graph has a Hamilton path in 1969. There is a growing interest in solving this longstanding problem and still it remains widely open. In fact, it was known that only five vertex-transitive graphs exist without a Hamiltonian cycle which do not belong to Cayley graphs. A Cayley graph is the subclass of vertex-transitive graph, and in view of the Lovász conjecture, the attention has focused more toward the Hamiltonicity of Cayley graphs. This survey will describe the current status of the search for Hamiltonian cycles and paths in Cayley graphs and digraphs on different groups, and discuss the future direction regarding famous conjecture.
A crucial practical issue encountered in logistics management is the circulation of final products from depots to end-user customers. When routing and scheduling systems are improved, they will not only improve customer satisfaction but also increase the capacity to serve a large number of customers minimizing time. On the assumption that there is only one depot, the key issue of distribution is generally identified and formulated as VRP standing for Vehicle Routing Problem. In case, a company having more than one depot, the suggested VRP is most unlikely to work out. In view of resolving this limitation and proposing alternatives, VRP with multiple depots and multi-depot MDVRP have been a focus of this paper. Carrying out a comprehensive analytical literature survey of past ten years on cost-effective Multi-Depot Vehicle Routing is the main aim of this research. Therefore, the current status of the MDVRP along with its future developments is reviewed at length in the paper.
This paper presents the design and construction of a faculty course timetable. The system uses an Integer Linear Programming model which attempts to assign groups of course units to time periods where each group is a result of a graph coloring approach. Limited number of lecture halls, large number of subject combinations and growing number of student registration have made the problem very tight which results thousands of variables and constraints to the model. The quality of the solution depends on the location of the time period assigned to the set of course units. Hence the objective function is defined to optimize the allocation of time periods to course units. The model results a feasible solution which has reduced the maximum idle time of students to three hours and it can be implemented with the lecture halls currently available in the faculty of Applied Sciences, University of Sri Jayewardenepura. The model is flexible and allows to change the constraints depending on the faculty requirements and other factors and if necessary, construct alternative schedules.
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