Covering arrays on graphs

  • Authors:

  • Karen Meagher;Brett Stevens

  • Affiliations:

  • Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada, ON;School of Mathematics and Statistics, Carleton University, Ottawa, Canada, ON

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2005

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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) 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) 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.