Covering arrays on graphs: Qualitative independence graphs and extremal set partition theory

Abstract

The main focus of this thesis is a generalization of covering arrays, covering arrays on graphs. Two vectors v,w in Z_k^n are qualitatively independent if for all ordered pairs (a,b) in Z_k x Z_k there is a position i in the vectors where (a,b) = (v_i,w_i). 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. 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 Erdos-Ko-Rado Theorem. Since the rows of general covering arrays correspond to set partitions, we give extensions of Sperner's Theorem and the Erdos-Ko-Rado Theorem to set-partition systems. We also consider a subgraph of a general qualitative independence graph called the uniform qualitative independence graph. 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.