Covering arrays on graphs

Two vectors v , w in Z g n are qualitatively independent if for all pairs ( a , b ) ∈ Z g × Z g there is a position i in the vectors where ( a , b ) = ( v i , w i ) . A covering array on a graph G, CA ( n , G , g ) , is a | V ( G ) | × n array on Z g 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 ( K ω ( G ) , g ) ⩽ CAN ( G , g ) ⩽ CAN ( K χ ( G ) , g ) . The question we ask is, are there graphs with CAN ( G , g ) < CAN ( K χ ( 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 ( K χ ( 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 1ʹs.