Double commutative-step digraphs with minimum diameters

From a natural generalization to E* of the concept of congruence, it is possible to define a family of 2-regular digraphs that we call double commutative-step digraphs. They turn out to be Cayley diagrams of Abelian groups generated by two elements, and can be represented by L-shaped tiles which tessellate the plane periodically. A double commutative-step digraph with n vertices is said to be tight if its diameter attains the lower bound Ib(n)=r&l-2. In this paper, the tiles associated with tight double commutative-step digraphs are fully characterized. This allows us to construct such digraphs and also to find for which values of II they exist. In particular, we characterize a complete set of families of tight double fixed-step digraphs (also called circulant digraphs), generalizing some previously known results.