| This thesis shows the existence of integer partitions in different fields of mathematics. We look at various problems involving integer partitions including its role in graph theory and lattice theory.;The involvement of integer partitions in the field of lattice theory is also discussed in this thesis. We take a look at a parallel cutting problem introduced by Ginsburg and Sands. This problem entails finding an optimal algorithm for breaking apart a set of integer length sticks into unit length sticks. We present a stronger optimal algorithm than the one proposed by Ginsburg and Sands. We then are brought to the idea of ordering of partitions and the Lattice of Integer Partitions. Finally, we relate the lattice obtained from the parallel cutting problem to the Lattice of Integer Partitions and study some extraordinary properties of this lattice.;We see how integer partitions are used in graph theory by investigating different graph decomposition problems. We present a proof for a problem involving the decomposition of paths into matchings of specified sizes. As well, we illustrate a proof for a known problem concerning the decomposition of complete graphs into matchings of given sizes. Furthermore, we investigate other graph decomposition problems including path and cycle decompositions of complete graphs. The problem of decomposing complete graphs into cycles of fixed or arbitrary lengths is discussed a great deal in this thesis. We see why the Alspach conjecture on arbitrary cycle lengths has brought immense interest to the area of cycle decomposition. |
