Kautz graph



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 $$s_0s_1 \cdots s_{N + 1}$$, 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 and fault-tolerant computing applications: such a network is known as a Kautz network.