On Cyclic Kautz Digraphs

TL;DR

This paper introduces modified cyclic Kautz digraphs that achieve minimal diameter for given vertices and out-degree, extending Kautz digraph properties with new cyclic constraints and regularity features.

Contribution

It defines cyclic Kautz digraphs with cyclic label properties, derives their parameters, and constructs modified versions with minimal diameter and regularity.

Findings

Defined cyclic Kautz digraphs with new cyclic label conditions.

Calculated key parameters: vertices, arcs, diameter.

Constructed modified digraphs with minimal diameter and regularity.

Abstract

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 $MCK(d,\ell)$ and it is derived from the Kautz digraphs $K(d,\ell)$. %It is not common to find non-regular digraphs with minimal diameter given their number of vertices and out-degree. It is well-known that the Kautz digraphs $K(d,\ell)$ have the smallest diameter among all digraphs with their number of vertices and degree. We define the cyclic Kautz digraphs $CK(d,\ell)$, whose vertices are labeled by all possible sequences $a_1\ldots a_\ell$ of length $\ell$, 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 $a_1\neq a_\ell$. The cyclic Kautz digraphs $CK(d,\ell)$ have arcs between vertices $a_1 a_2\ldots a_\ell$ and $a_2 \ldots a_\ell a_{\ell+1}$, with $a_1\neq a_\ell$ and $a_2\neq a_{\ell+1}$. Unlike in Kautz digraphs $K(d,\ell)$, any label of a vertex of $CK(d,\ell)$ can be cyclically shifted to form again a label of a vertex of $CK(d,\ell)$. We give the main parameters of $CK(d,\ell)$: number of vertices, number of arcs, and diameter. Moreover, we construct the modified cyclic Kautz digraphs $MCK(d,\ell)$ to obtain the same diameter as in the Kautz digraphs, and we show that $MCK(d,\ell)$ are $d$-out-regular. Finally, we compute the number of vertices of the iterated line digraphs of $CK(d,\ell)$.