  It is proved that 1) the indicator function of some onefold or multifold
independent set in a regular graph is a perfect coloring if and only if the set
attain the Delsarte--Hoffman bound; 2) each transversal in a uniform regular
hypergraph is an independent set attaining the Delsarte--Hoffman bound in the
vertex adjacency multigraph of this hypergraph; 3) combinatorial designs with
parameters $t$-$(v,k,\lambda)$ and similar $q$-designs, difference sets,
Hadamard matrices, and bent functions are equivalent to perfect colorings of
special graphs and multigraphs, in particular, it is true in the cases of the
Johnson graphs $J(n,k)$ for $(k-1)$-$(v,k,\lambda)$ designs and the Grassmann
graphs $J_2(n,2)$ for bent functions.
  Keywords: perfect coloring, equitable partition, transversal of hypergraph,
combinatorial design, $q$-design, difference set, bent function, Johnson graph,
Grassmann graph, Delsarte--Hoffman bound
