From Subkautz Digraphs to Cyclic Kautz Digraphs

The Kautz digraphs K(d, ℓ) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related to these, the cyclic Kautz digraphs CK(d, ℓ) were recently introduced by Böhmová, Huemer and the author, and some of its distance-related parameters were fixed. In this paper we propose a new approach to the cyclic Kautz digraphs by introducing the family of the subKautz digraphs sK(d, ℓ), from where the cyclic Kautz digraphs can be obtained as line digraphs. This allows us to give exact formulas for the distance between any two vertices of both sK(d, ℓ) and CK(d, ℓ). Moreover, we compute the diameter and the semigirth of both families, also providing efficient routing algorithms to find the shortest path between any pair of vertices. Using these parameters, we also prove that sK(d, ℓ) and CK(d, ℓ) are maximally vertex-connected and super-edge-connected. Whereas K(d, ℓ) are optimal with respect to the diameter, we show that sK(d, ℓ) and CK(d, ℓ) are optimal with respect to the mean distance, whose exact values are given for both families when ℓ = 3. Finally, we provide a lower bound on the girth of CK(d, ℓ) and sK(d, ℓ).

The diameter of cyclic Kautz digraphs

We present a new kind of digraphs, called cyclic Kautz digraphs CK(d,l), which are subdigraphs of the well-known Kautz digraphs K(d,l). The latter have the smallest diameter among all digraphs with their number of vertices and degree. Cyclic Kautz digraphs CK(d,l) have vertices labeled by all possible sequences a1...al of length l, such that each character ai 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 a1 ? al. Their arcs are between vertices a1a2... al and a2... alal+1, with a1 ? al and a2 ? al+1. We give the diameter of CK(d,l) for all the values of d and l, and also its number of vertices and arcs.