The term "
-crossed
prism graph" is used in this work for the graph obtained
from two disjoint cycle
graphs 
(for even 
), with vertices 
, ..., 
and 
, ..., 
in cyclic order, by adding edges 
and 
for 
, 3, ..., 
.
The crossed prism graphs are cubic vertex-transitive (and hence appear in Read and Wilson 1998, though without any designation indicating
membership in a special graph family), weakly
regular, Hamiltonian, and Hamilton-laceable.
The 
-crossed
prism graphs are toroidal for 
(E. Weisstein, May 9, 2023).
Simmons (2014) used the term "polygonal bigraph on 
vertices" for graphs isomorphic
to the 
-crossed
prism graph and investigated the Hamilton-laceability
and structure of Hamiltonian paths in these graphs.
The cases 
and 6 have graph crossing number 0 and 2,
respectively. For every even 
, drawing the two cycles
as concentric circles and routing each pair
of crossed joining edges within a separate sector of
the annulus gives exactly 
crossings, proving the upper
bound 
.
The upper bound is known to be attained for every
even 
with 
and is conjectured
to be attained for all even 
.
The first few crossed prism graphs and some of their properties are implemented in the Wolfram Language as GraphData[
"CrossedPrism", n
].
The 
-crossed
prism graph has independence polynomial
![I_n(x)=2^(-n)[(1+2x(2+x)-sqrt((1+2x)(1+6x)))^n+(1+2x(2+x)+sqrt((1+2x)(1+6x)))^n],](/images/equations/CrossedPrismGraph/NumberedEquation1.svg) |
which has recurrence equation
 |
The 
-crossed
prism graph is isomorphic to the Haar graph 
and to the 
-honeycomb
toroidal graph. The 8-crossed prism graph is isomorphic to the 
truncated
square lattice graph, illustrated above. Other special cases are summarized in
the following table.
 |  -crossed prism graph |
| 4 | cubical graph |
| 6 | Franklin graph |
| 8 |  truncated
square lattice graph, cubic vertex-transitive
graph Ct19 |
| 10 | cubic vertex-transitive graph Ct29 |
| 12 | cubic vertex-transitive graph Ct42 |
| 14 | cubic vertex-transitive graph Ct54 |
| 16 | cubic vertex-transitive graph Ct74 |
