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

Two vectors v, w in Zgn are qualitatively independent if for all pairs (a, b) ∈ Zg × Zg there is a position i in the vectors where (a, b) = (vi, wi). A covering array on a graph G, CA (n, G, g), is a |V(G)| × n array on Zg with the property that any two rows which correspond to adjacent vertices in G are qualitatively independent. The smallest possible n is denoted by CAN(G, g). These are an extension of covering arrays. It is known that CAN(Kω(G), g) ≤ CAN(G, g) ≤ CAN(Kχ(G), g). The question we ask is, are there graphs with CAN(G, g) < CAN(Kχ(G), g)? We find an infinite family of graphs that satisfy this inequality. Further we define a family of graphs QI(n, g) that have the property that there exists a CAN(n, G, g) if and only if there is a homomorphism to QI(n, g). Hence, the family of graphs QI(n, g) defines a generalized colouring. For QI(n, 2), we find a formula for both the chromatic and clique number and determine two necessary conditions for CAN (G, 2) < CAN(Kχ(G), 2). We also find the cores of all the QI(n, 2) and use this to prove that the rows of any covering array with g = 2 can be assumed to have the same number of 1's.