Chiniforooshan, Ehsan Jay
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In this thesis, we introduce the intersperse coloring problem, which is a generalized version of the hypergraph coloring problem. In the intersperse coloring problem, we seek a coloring that assigns at least l different colors to each hyperedge of the input hypergraph, where l is an input parameter of the problem. We show that the notion of intersperse coloring unifies several well-known coloring problems, in addition to the conventional graph and hypergraph coloring problems, such as the strong coloring of hypergraphs, the star coloring problem, the problem of proper coloring of graph powers, the acyclic coloring problem, and the frugal coloring problem. We also provide a number of upper and lower bounds on the intersperse coloring problem on hypergraphs in the general case. The nice thing about our general bounds is that they can be applied to all the coloring problems that are special cases of the intersperse coloring problem. In this thesis, we also propose a new model for graph and hypergraph property testing, called the symmetric model. The symmetric model is the first model that can be used for developing property testing algorithms for non-uniform hypergraphs. We also prove that there exist graph properties that have efficient property testers in the symmetric model but do not have any efficient property tester in previously-known property testing models.