4 research outputs found
The genus of generalized random and quasirandom graphs
The genus of a graph is the minimum integer such that has an embedding in some surface (closed compact 2-manifold) of genus . In this thesis, we will discuss the genus of generalized random and quasirandom graphs. First, by developing a general notion of random graphs, we determine the genus of generalized random graphs. Next, we approximate the genus of dense generalized quasirandom graphs. Based on analysis of minimum genus embeddings of quasirandom graphs, we provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. More precisely, we provide an algorithm that for a given (dense) graph of order and given varepsilon>0, returns an integer such that has an embedding into a surface of genus , and this is -close to a minimum genus embedding in the sense that the minimum genus of satisfies: . The running time of the algorithm is , where is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) of genus . This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time . The last part of the thesis studies the genus of complete -uniform hypergraphs, which is a special case of genus of random bipartite graphs, and also a natural generalization of Ringel--Youngs Theorem. Embeddings of a hypergraph are defined as the embeddings of its associated Levi graph with vertex set , in which and are adjacent if and only if and are incident in . The construction in the proof may be of independent interest as a design-type problem