Combinatorial designs, difference sets and bent functions as perfect colorings of graphs and multigraphs

Abstract: 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