Covering Arrays on Graphs: Qualitative Independence Graphs and Extremal Set Partition Theory

There has been a good deal of research on covering arrays over the last 20 years. Most of this work has focused on constructions, applications and generalizations of covering arrays. The main focus of this thesis is a generalization of covering arrays, covering arrays on graphs. The original motivation for this generalization was to improve applications of covering arrays to testing systems and networks, but this extension also gives us new ways to study covering arrays. Two vectors v, w in Zk are qualitatively independent if for all ordered pairs (a, b) ∈ Zk × Zk there is a position i in the vectors where (a, b) = (vi, wi). A covering array is an array with the property that any pair of rows are qualitatively independent. A covering array on a graph is an array with a row for each vertex of the graph with the property that any two rows which correspond to adjacent vertices are qualitatively independent. A covering array on the complete graph is a covering array. A covering array is optimal if it has the minimum number of columns among covering arrays with the same number of rows. The addition of a graph structure to covering arrays makes it possible to use methods from graph theory to study these designs. In this thesis, we define a family of graphs called the qualitative independence graphs. A graph has a covering array, with given parameters, if and only if there is a homomorphism from the graph to a particular qualitative independence graph. Cliques in qualitative independence graphs relate to covering arrays and independent sets are connected to intersecting partition systems. It is known that the exact size of an optimal binary covering array can be determined using Sperner’s Theorem and the Erdős-Ko-Rado Theorem. In this thesis, we find good bounds on the size of an optimal binary covering array on a graph. In addition, we determine both the chromatic number and a core of the binary qualitative independence graphs. Since the rows of general covering arrays correspond to set partitions, we give extensions of Sperner’s Theorem and the Erdős-Ko-Rado Theorem to set-partition systems. These results are part of a general framework to study extremal partition systems. The core of the binary qualitative independence graphs can be generalized to a subgraph of a general qualitative independence graph called the uniform qualitative independence graph. Cliques in the uniform qualitative independence graphs relate to balanced covering arrays. Using these graphs, we find bounds on the size of a balanced covering array. We give the spectra for several of these graphs and conjecture that they are graphs in an association scheme. We also give a new construction for covering arrays which yields many new upper bounds on the size of optimal covering arrays.