Minimal harmoniously colorable designs

A graph G is minimal harmoniously colorable if it has a proper vertex coloring in which each pair of colors occurs exactly once on an edge. In particular, if D is a 2‐design we consider the graph G whose vertices are the points and blocks of D and where two vertices of G are adjacent if and only if the corresponding elements of D are incident. It will be shown that if D is symmetric then G is minimal harmoniously colorable if and only if D is a Hadamard design with corresponding Hadamard matrix of a certain form. We obtain some results if D is nonsymmetric, and construct two classes of nonsymmetric minimal harmoniously colorable designs. © 1994 John Wiley & Sons, Inc.