An infinite family of self-diclique digraphs

Let D = ( V , A ) be a digraph. Consider X and Y (not necessarily disjoint), nonempty subsets of vertices of D . We define a disimplex K ( X , Y ) of D to be the subdigraph whose vertex set is V ( K ( X , Y ) ) = X ∪ Y and in which an arc goes from every vertex of X to every vertex of Y (when X ∩ Y ≠ ∅ , loops are not considered). A disimplex K ( X , Y ) is called a diclique of D if K ( X , Y ) is not a proper subdigraph of any other disimplex of D . The diclique digraph (or diclique operator) k ⃗ ( D ) of a digraph D is the digraph whose vertex set is the set of all dicliques of D and ( K ( X , Y ) , K ( X ′ , Y ′ ) ) is an arc of k ⃗ ( D ) if and only if Y ∩ X ′ ≠ ∅ . We say that a digraph D is self-diclique if k ⃗ ( D ) is isomorphic to D . In this paper we exhibit an infinite family of self-diclique circulant digraphs for which one of its members is an Eulerian orientation of the graph of the regular octahedron. This family is a natural generalization of the example given in Zelinka (2002) [5].