Topics in graph decompositions and related algebras

Robert Brier (2008). Topics in graph decompositions and related algebras PhD Thesis, School of Physical Sciences, The University of Queensland.

       
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Author Robert Brier
Thesis Title Topics in graph decompositions and related algebras
School, Centre or Institute School of Physical Sciences
Institution The University of Queensland
Publication date 2008-06
Thesis type PhD Thesis
Supervisor Bryant, Darryn E.
Maenhaut, Barbara
Subjects 230000 Mathematical Sciences
Formatted abstract The majority of this thesis is an investigation of algebras arising from K-perfect m-cycle systems. The first two chapters contain combinatorial results, while the next two give the algebraic results.
The first chapter contains results on edge-coloured graph decompositions. Using a re¬sult from [23], we find necessary and sufficient conditions for the existence of a non-trivial complete edge-coloured graph which has a decomposition into ‘isomorphic’ copies of a given edge-coloured graph. A sufficient condition is also given for ensuring the embedding of partial decompositions in complete decompositions of finite order.
In the second chapter, K-perfect m-circuit systems and K-perfect m-cycle systems are introduced. Results from the second chapter show the existence of K-perfect m-cycle sys¬tems, and also of a particular class of K-perfect m-circuit systems, for general K and m.It is also shown that every partial K-perfect m-cycle system embeds in a K-perfect m-cycle system of finite order. The proofs of these results are based on results in the first chapter.
The third chapter discusses a type of algebra that arises from a K-perfect m-cycle system. These algebras generalise the Steiner quasigroups which arise from Steiner triple systems (that is 3-cycle systems). It is determined when the class of algebras arising from the K-perfect m-cycle systems is a variety. This generalises the results of [9] and [13].
The fourth chapter continues investigation of the variety generated by algebras arising from K-perfect m-cycle systems. It is shown that the free algebras of this variety arise from cycle systems. It also shown that the algebras arising from finite K-perfect m-cycle systems, and those arising from finite K-perfect m-cycle systems with K� ⊆ K, generate the same
variety. Two questions asked in [10] are also answered.
In the fifth chapter we introduce the problem of determining the parameters for which there exist two arrays, each row-Latin and column-Latin, which can be superimposed to obtain every 2-element subset of an n-set exactly once. A complete solution in the case that n is odd is given. There are also partial results for the case of even n, and discussion and results on generalisations of this problem.



 
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