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

V. N. Potapov, S. V. Avgustinovich

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 $t$-$(v,k,λ)$ 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 $(k-1)$-$(v,k,λ)$ 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

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 - and similar -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 for - designs and the Grassmann graphs for bent functions. Keywords: perfect coloring, equitable partition, transversal of hypergraph, combinatorial design, -design, difference set, bent function, Johnson graph, Grassmann graph, Delsarte--Hoffman bound
Paper Structure (6 sections, 20 theorems, 4 equations)

This paper contains 6 sections, 20 theorems, 4 equations.

Table of Contents

  1. Perfect Colorings of Multigraphs and Hypergraphs
  2. Transversals of Hypergraphs
  3. Independent Sets
  4. Combinatorial Designs
  5. Difference Sets and Bent Functions
  6. Difference Sets, Strongly Regular Graphs and Bent Functions as Perfect Colorings

Key Result

Proposition 1

$1.$ The eigenvalues of $\mathcal{M}_{12}(G)$ are nonnegative. $2.$ The restriction of every eigenfunction of $G$ with eigenvalue $\theta$ to the first part of the graph is an eigenfunction of $\mathcal{M}_{12}(G)$ with the eigenvalue $\theta^2$. $3.$ Every eigenfunction of $\mathcal{M}_{12}(G)$ wit

Theorems & Definitions (20)

  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Corollary 2
  • Proposition 3
  • Corollary 3
  • Proposition 4
  • Proposition 5
  • Theorem 1
  • Proposition 6
  • ...and 10 more