Covering arrays on graphs

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)