COMBINATORIAL DESIGNS, DIFFERENCE SETS, AND BENT FUNCTIONS AS PERFECT COLORINGS OF GRAPHS AND MULTIGRAPHS

We prove that (1): the characteristic function of each independent set in each regular graph attaining the Delsarte-Hoffman bound is a perfect coloring; (2): each transversal in a uniform regular hypergraph is an independent set in the vertex adjacency multigraph of a hypergraph attaining the Delsarte-Hoffman bound for this multigraph; and (3): the combinatorial designs with parameters t-(v, k, lambda) and their q-analogs, difference sets, Hadamard matrices, and bent functions are equivalent to perfect colorings of some graphs of multigraphs, in particular, the Johnson graph J(n, k) for (k - 1) - (v, k, lambda)-designs and the Grassmann graph J(2)(n, 2) for bent functions.