Covering arrays on graphs

Abstract

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.