| Summary: | A central problem in topological graph theory is determining the (orientable or nonorientable) genus of a given graph <i>G</i>. For a general graph <i>G</i>, this problem is very difficult (in fact, it is NP-complete). Thus, it is desirable to have large families of graphs for which the genus is known. In this thesis, we use hamilton cycle embeddings of complete tripartite graphs to help determine the genera of two special families of graphs.
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The first family of graphs -- and the original motivation for this problem -- is the collection of join graphs
<i>F</i><sub>1</sub> = {<span style="text-decoration: overline"><i>K<sub>m</sub></i></span>+<i>G</i> | <i>G</i> is an <i>n</i>-vertex graph and <i>m</i> ≥ <i>n</i> − 1}.
The nonorientable genus for this family of graphs has been completely determined by Ellingham and Stephens. In the orientable case, they were able to determine the genus for some infinite subfamilies of <i>F</i><sub>1</sub>, but their results are not complete. In this thesis, we present a tripling construction for these embeddings that is an extension of a doubling construction employed by Ellingham and Stephens. Using this construction, together with hamilton cycle embeddings of complete tripartite graphs, we obtain the orientable genus for some additional infinite subfamilies of <i>F</i><sub>1</sub>.
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The second family of graphs considered is the collection of complete quadripartite graphs
<i>F</i><sub>2</sub> = {<i>K<sub>t,n,n,n</sub></i> | <i>t</i> ≥ 2<i>n</i>}. Using hamilton cycle embeddings of <i>K<sub>n,n,n</sub></i> and some surgical techniques, we determine the orientable and nonorientable genus for all members of <i>F</i><sub>2</sub>.
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We use two main methods to build the hamilton cycle embeddings of complete tripartite graphs that are required to obtain these results. The first method is a cyclic construction that generates the entire embedding from a sequence of edge slopes that satisfies certain conditions. Using this construction and some lifting results due to Bouchet and his collaborators, we obtain nonorientable hamilton cycle embeddings of <i>K<sub>n,n,n</sub></i> for all <i>n</i> ≥ 2. The second method is a construction that utilizes a connection to orthogonal latin squares that exhibit some additional structure. Using this construction -- and filling in the case <i>n</i> = 2<i>p</i> for a prime <i>p</i> with voltage graphs -- we obtain orientable hamilton cycle embeddings of <i>K<sub>n,n,n</sub></i> for all <i>n</i> ≠ 2.
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