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- W2116737272 abstract "In this thesis we investigate perfect sets of Euler tours of complete graphs Kn and Hamilton decompositions of the line graphs of complete graphs L . We also present some partial results in the area of pairwise compatible Hamilton path decompositions of the graph hK. and pairwise compatible Hamilton decompositions of the graph K 2 k + l . Chapter 1 contains definitions and notation, and an introduction that outlines some of the work that has been done in the areas of pairmise compatible Euler tours of graphs, Hamilton decompositions of L(h' , ) , and Dudeney sets. We also present the problems that will be considered iri the thesis. Eiotzig conjectured in 19'79 that has a perfect set of Euler tours for all positive integers k. In Chapter 2 we give a constructive proof of his conjecture. McKay conjectured that L(Kn) has a Hamilton decomposition for all n. When n is odd, this conjecture is a corollary of Kotzig's conjecture. In Chapter 3 we consider one n-ay in which we could extend the definition of a perfect set of Euler tours to include IGk. a graph that has no Euler tour. Since our goal is to have a Hamilton decomposition of L ( h k ) as a corollary. we define a perfect set of Euler tours of K2k f I. where I is a 1-factor of J t ; k . to be a set of Euler tours of h;& + I such that every ?-path of KZk is in exactly one of the tours and such that for every edge a b E I. each qf the Euler tours either uses the digon a ba or the digon ba 6. We then give a constructive proof of a perfect set of Euler tours of + I, and = * thereby give a cornpiexion of the proof oi ~1lcECay's conjecture. The results in Chapter 4 were motivated by another question of Kotzig's: What is the smallest k for which there is a perfect set of Hamilton docompositions of KZkS1? -* * We prove for aii k > I that has at ieast 2k '2 pairwise compatible &milton path decompositions. This is one less than the maximum possible of 2k 1. In the case of K4, it is straightforward to show it is best possible. ?7e then construct a set of 411: 2 Hamilton path decompositions of that between them contain every 2-pat h of the graph exac t l~ twice. ?Ye also find a lower bound on t-he number of pairwise compatible Hamilton decompositions of Vlk present our concIusions in Chapter 5." @default.
- W2116737272 created "2016-06-24" @default.
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- W2116737272 date "1996-01-01" @default.
- W2116737272 modified "2023-09-27" @default.
- W2116737272 title "Perfect sets of Euler tours of complete graphs" @default.
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