A KC graph is the graph Cartesian product 
of a complete
graph 
and a cycle graph 
. Knuth (2025, p. 22) uses this name for parameters
and 
. The KC graphs are regular
with vertex degree 
and have vertex count and
edge count
 |  |  |
(1)
|
 |  |  |
(2)
|
Many KC graphs are circulant graphs. In particular, for any 
(i.e., relatively
prime so as to have no common divisor greater than
1), 
is isomorphic
to the circulant graph 
, where the indices
are a subset of integer multiples of 
and 
less than or equal to 
.
Other special cases are summarized in the table below.
| graph | special case |
 | cycle graph  |
 | prism
graph  |
 | rook graph  |
 | generalized
quadrangle  |
for  ,  |  torus grid graph |
The graph crossing number of 
is 
for integer 
(Beineke and Ringeisen 1980).
is ungraceful
for 
odd (Knuth 2025, p. 22).
The scramble number of the KC graph 
with 
and 
is 
(Echavarria et al. 2021).
