A complete tripartite graph is the 
case of a complete
k-partite graph. In other words, it is a tripartite graph (i.e., a set
of graph vertices decomposed into three disjoint
sets such that no two graph vertices within the same
set are adjacent) such that every vertex of each set graph
vertices is adjacent to every vertex in the other two sets. If there are 
,
,
and 
graph vertices in the three sets, the complete tripartite
graph (sometimes also called a complete trigraph) is denoted 
.
Special classes are summarized in the following table.
 | special class |
 |  -fan graph |
 |  -fan graph |
 |  -cone graph |
Some special cases are summarized in the following table, some of which are illustrated above.
 | name |  |
| 3 | triangle graph |  |
| 4 | diamond graph |  |
| 5 | (3,2)-fan graph |  |
| 5 | 5-wheel graph |  |
| 6 | octahedral graph |  |
| 6 | (3,3)-fan graph |  |
| 6 | (4,2)-fan graph |  |
| 7 | (4,3)-cone graph |  |
| 7 | (4,3)-fan graph |  |
| 7 | (5,2)-fan graph |  |
| 8 | (4,4)-cone graph |  |
| 8 | (5,3)-fan graph |  |
| 9 | 9-circulant graph  |  |
| 9 | (4,5)-cone graph |  |
| 12 | circulant
graph  |  |
| 15 | circulant
graph  |  |
The number of Hamiltonian cycles in the graph 
can be efficiently computed using a general complete
k-partite graph recurrence (Horák and Tovarek 1979). A closed-form
formula is also known (submitted as a problem, 2025).
-Partite Graphs." Math. Slovaca 29, 43-47,
1979.