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a b s t r a c tWe prove that the complete graph with a hole K u+w − K u can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, each cycle has length at most min(u, w), and the longest cycle is at most three times as long as the second longest. This generalises existing results on decomposing the complete graph with a hole into cycles of uniform length, and complements work on decomposing complete graphs, complete multigraphs, and complete multipartite graphs into cycles of arbitrary specified lengths.

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We prove that the complete graph with a hole K u+w − K u can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, each cycle has length at most min(u, w), and the longest cycle is at most three times as long as the second longest. This generalises existing results on decomposing the complete graph with a hole into cycles of uniform length, and complements work on decomposing complete graphs, complete multigraphs, and complete multipartite graphs into cycles of arbitrary specified lengths.

In 1973, Doyen and Wilson [7] famously solved the problem of when a 3-cycle system can be embedded in another 3-cycle system. There has been much interest in the literature in generalising this result for m-cycle systems when m > 3. Although there are several partial results, including complete solutions for some small values of m and strong partial results for even m, this still remains an open problem [4,5,8,9].The main results of this thesis concern generalisations of the Doyen-Wilson theorem for odd m-cycle systems and cycle decompositions of the complete graph with a hole. The complete graph of order v with a hole of size u, K v − K u , is constructed from the complete graph of order v by removing the edges of a complete subgraph of order u (where v ≥ u).For each odd m ≥ 3 we completely solve the problem of when an m-cycle system of order u can be embedded in an m-cycle system of order v, barring a finite number of possible exceptions. The problem is completely resolved in cases where u is large compared to m, where m is a prime power, or where m ≤ 15. In other cases, the only possible exceptions occur when v − u is small compared to m. This result is proved as a consequence of a more general result which gives necessary and sufficient conditions for the existence of an m-cycle decomposition of K v − K u in the case where u ≥ m − 2 and v − u ≥ m + 1 both hold.We prove that K v − K u can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, v − u ≥ 10, each cycle has length at most min (u, v − u), and the longest cycle is at most three times as long as the second longest. This complements existing results for cycle decompositions of graphs such as the complete graph [1,3,10], complete bipartite graph [6,8] and complete multigraph [2].