Kautz graph

Example of Kautz graph on 3 characters with string length 2 (on the left) and 3 (on the right); the edges on the left correspond to the vertices on the right.

The Kautz graph $K_{M}^{{N+1}}$ is a directed graph of degree $M$ and dimension $N+1$ , which has $(M+1)M^{{N}}$ vertices labeled by all possible strings $s_{0}\cdots s_{N}$ of length $N+1$ which are composed of characters $s_{i}$ chosen from an alphabet $A$ containing $M+1$ distinct symbols, subject to the condition that adjacent characters in the string cannot be equal ( $s_{i}\neq s_{{i+1}}$ ).

The Kautz graph $K_{M}^{{N+1}}$ has $(M+1)M^{{N+1}}$ edges

$\{(s_{0}s_{1}\cdots s_{N},s_{1}s_{2}\cdots s_{N}s_{{N+1}})|\;s_{i}\in A\;s_{i}\neq s_{{i+1}}\}\,$

It is natural to label each such edge of $K_{M}^{{N+1}}$ as $s_{0}s_{1}\cdots s_{{N+1}}$ , giving a one-to-one correspondence between edges of the Kautz graph $K_{M}^{{N+1}}$ and vertices of the Kautz graph $K_{M}^{{N+2}}$ .

Kautz graphs are closely related to De Bruijn graphs.

Properties

For a fixed degree $M$ and number of vertices $V=(M+1)M^{N}$ , the Kautz graph has the smallest diameter of any possible directed graph with $V$ vertices and degree $M$ .
All Kautz graphs have Eulerian cycles. (An Eulerian cycle is one which visits each edge exactly once—This result follows because Kautz graphs have in-degree equal to out-degree for each node)
All Kautz graphs have a Hamiltonian cycle (This result follows from the correspondence described above between edges of the Kautz graph $K_{M}^{{N}}$ and vertices of the Kautz graph $K_{M}^{{N+1}}$ ; a Hamiltonian cycle on $K_{M}^{{N+1}}$ is given by an Eulerian cycle on $K_{M}^{{N}}$ )
A degree- $k$ Kautz graph has $k$ disjoint paths from any node $x$ to any other node $y$ .

In computing

The Kautz graph has been used as a network topology for connecting processors in high-performance computing^[1] and fault-tolerant computing^[2] applications: such a network is known as a Kautz network.

Notes

↑ Darcy, Jeff (2007-12-31). "The Kautz Graph". Canned Platypus. External link in |publisher= (help)
↑ Li, Dongsheng; Xicheng Lu; Jinshu Su (2004). "Graph-Theoretic Analysis of Kautz Topology and DHT Schemes". Network and Parallel Computing: IFIP International Conference. Wuhan, China: NPC. pp. 308–315. ISBN 3-540-23388-1. Retrieved 2008-03-05.

This article incorporates material from Kautz graph on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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