Generalisations of the Doyen-Wilson Theorem
HoyteRosalind A.
2017
In 1973, Doyen and
Wilson 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 <i>m</i>-cycle systems when <i>m</i> > 3. Although
there are several partial results, including complete solutions for some small
values of <i>m</i> and strong partial results for even <i>m</i>, this still remains an open
problem. <br>
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The main results of this thesis concern generalisations of
the Doyen-Wilson Theorem for odd <i>m</i>-cycle systems and cycle decompositions of
the complete graph with a hole. The complete graph of order <i>v</i> with a hole of
size <i>u</i>, <i>K</i><sub>v </sub>-<i> K</i><sub>u,</sub> is constructed from the complete graph of order <i>v</i> by removing
the edges of a complete subgraph of order <i>u</i> (where <i>v </i>≥<i> u</i>). <br>
<br>
For each odd <i>m</i> ≥ 3 we completely solve the problem of when an
<i>m</i>-cycle system of order <i>u</i> can be embedded in an m-cycle system of order <i>v</i>,
barring a finite number of possible exceptions. The problem is completely
resolved in cases where <i>u</i> is large compared to <i>m</i>, where <i>m</i> is a prime power, or
where <i>m </i>≤ 15. In other cases, the only possible exceptions occur when <i>v</i> - <i>u</i> is
small compared to <i>m</i>. This result is proved as a consequence of a more general
result which gives necessary and sufficient conditions for the existence of an
<i>m</i>-cycle decomposition of <i>K</i><sub>v </sub><i>- K<sub>u</sub></i> in the case where <i>u </i>≥ <i>m</i> - 2 and <i>v</i> -<i> u</i> ≥<i> m</i> + 1 both hold.
<br>
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We prove that <i>K</i><sub>v </sub><i>- K<sub>u</sub></i> can be decomposed into cycles of
arbitrary specified lengths provided that the obvious necessary conditions are
satisfied, <i>v </i>-<i> u</i> ≥ 10, each cycle has length at most min(<i>u,v - u</i>), 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,
complete bipartite graph and complete multigraph. <br>
<br>
We obtain these cycle decomposition results by applying a
cycle switching technique to modify cycle packings of <i>K<sub>v</sub> - K<sub>u</sub></i>. The tools
developed by cycle switching enable us to merge collections of short cycles to
obtain longer cycles. The methodology therefore relies on first finding
decompositions of various graphs into short cycles, then applying the merging
results to obtain the required decomposition. Similar techniques have
previously been successfully applied to the complete graph and the complete
bipartite graph. These methods also have potential to be further developed for
the complete graph with a hole as well as other graphs. <br>
<br>
We also give a complete solution to the problem of when there
exists a packing of the complete multigraph with cycles of arbitrary specified
lengths. The proof of this result relies on applying cycle switching to modify
cycle decompositions of the complete multigraph obtained from known results. <br>
<br>
The results in this thesis make substantial progress toward
generalising the Doyen-Wilson Theorem for arbitrary odd cycle systems and
toward constructing cycle decompositions of the complete graph with a hole.
However there still remain unsolved cases. Moreover, the cycle switching and
base decomposition methods used to obtain these results give rise to several
interesting open problems.