A Kautz graph is a regular directed graph derived from a de Bruijn graph on an
alphabet of 
letters using words of length 
by deleting words containing two or more consecutive
identical letters. For example, the 
-Kautz graph is illustrated above.
-Kautz graphs for 
to 4 and 
to 3 are illustrated above.
Kautz graphs are Eulerian and Hamiltonian. The 
-Kautz graph has vertex
count 
,
graph diameter 
, and vertex degree 
.
The Kautz graphs have the smallest possible graph diameter among all directed graphs for any fixed degree and number of vertices.
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