### On Cyclic Kautz Digraphs [article]

Katerina Böhmová, Cristina Dalfó, Clemens Huemer
2015 arXiv   pre-print
A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d,ℓ) and it is derived from the Kautz digraphs K(d,ℓ). non-regular digraphs with minimal diameter given their number of vertices and
more » ... t-degree. It is well-known that the Kautz digraphs K(d,ℓ) have the smallest diameter among all digraphs with their number of vertices and degree. We define the cyclic Kautz digraphs CK(d,ℓ), whose vertices are labeled by all possible sequences a_1... a_ℓ of length ℓ, such that each character a_i is chosen from an alphabet containing d+1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and now also requiring that a_1≠ a_ℓ. The cyclic Kautz digraphs CK(d,ℓ) have arcs between vertices a_1 a_2... a_ℓ and a_2 ... a_ℓ a_ℓ+1, with a_1≠ a_ℓ and a_2≠ a_ℓ+1. Unlike in Kautz digraphs K(d,ℓ), any label of a vertex of CK(d,ℓ) can be cyclically shifted to form again a label of a vertex of CK(d,ℓ). We give the main parameters of CK(d,ℓ): number of vertices, number of arcs, and diameter. Moreover, we construct the modified cyclic Kautz digraphs MCK(d,ℓ) to obtain the same diameter as in the Kautz digraphs, and we show that MCK(d,ℓ) are d-out-regular. Finally, we compute the number of vertices of the iterated line digraphs of CK(d,ℓ).