We present a new kind of digraphs, called cyclic Kautz digraphs CK(d,), which are subdigraphs of the well-known Kautz digraphs K(d,). The latter have the smallest diameter among all digraphs with their number of vertices and degree. Cyclic Kautz digraphs CK(d,) have vertices 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. Their arcs are between vertices a 1 a 2. .. a and a 2. .. a a +1 , with a 1 a and a 2 a +1. We give the diameter of CK(d,) for all the values of d and , and also its number of vertices and arcs.