### The diameter of cyclic Kautz digraphs

Katerina Böhmová, Cristina Dalfó, Clemens Huemer
2017 Filomat
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 degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d, ) and it is derived from the Kautz digraphs K(d, ). It is well-known that the Kautz digraphs K(d, ) have the smallest diameter among
more » ... digraphs with their number of vertices and degree. Here we define the cyclic Kautz digraphs CK(d, ), whose vertices are labeled by all possible sequences a 1 . . . a of length , 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 . The cyclic Kautz digraphs CK(d, ) have arcs between vertices a 1 a 2 . . . a and a 2 . . . a a +1 , with a 1 = a , a 2 = a +1 , and a i = a i+1 for i = 1, . . . , − 1. The cyclic Kautz digraphs CK(d, ) are subdigraphs of the Kautz digraphs K(d, ). We give the main parameters of CK(d, ) (number of vertices, number of arcs, and diameter). Moreover, we construct the modified cyclic Kautz digraphs MCK(d, ) to obtain the same diameter as in the Kautz digraphs, and we show that MCK(d, ) are d-out-regular. Finally, we compute the number of vertices of the iterated line digraphs of CK(d, ).