A twisted Isaacs graph is a cubic graph obtained by closing a chain of 
claw graphs with a permutation 
of their three leaves (Nedela and Škoviera
2022). For integer 
, take vertices 
, 
, 
, and 
for 
, ..., 
. Join 
to 
, 
, and 
, and join equally lettered vertices
at consecutive indices. Finally join 
, 
, and 
to 
, 
, and 
, respectively. The result has 
vertices and 
edges.
For each 
,
there are three non-isomorphic cases, denoted
by Fouquet et al. (2010). The parameter 
counts the cycles remaining
after the vertices 
are deleted, not the order of the closing permutation.
A 3-permutation cycle gives 
, with one remaining cycle
graph 
.
A transposition gives 
, with remaining cycle graphs 
and 
. The identity permutation
gives 
,
with three remaining cycle graphs 
. All three cases, including the identity
permutation, are called twisted Isaacs graphs by Nedela and Škoviera (2022).
The case 
is also called an Isaacs graph (Nedela and Škoviera 2022). Some named special
cases are summarized in the following table.
 | named graph |
 | triplex graph |
 | Tietze graph |
 | starfish graph |
 ,
odd  | flower
snark  |
Zheng et al. (2008, Lemmas 4.2, 4.3, and 4.8) determine the graph crossing number of 
as
 |
(1)
|
The abstract and introduction of Zheng et al. (2008) incorrectly print 
in the first case, giving 1, 2, and
3 for 
,
4, and 5. Lemmas 4.2 and 4.3 establish the correct values 2, 3, and 4.
For every integer 
, the graph crossing
number of 
is at most 
.
To see this, draw its three cycles as concentric
circles and place the claw graphs in separate angular
sectors, each with exactly one crossing of the middle cycle
graph. The graph crossing number of 
is known to equal 
at least for 
, 12, and 14.
The number of perfect matchings in all three cases is
 |
(2)
|
where 
,
,
and 
(Fouquet et al. 2010, Theorem 5). The three counts are distinct for each 
, which also proves that the three cases
are non-isomorphic.
All three cases have girth 6 and cyclic edge connectivity 6 for 
. For 
, both conclusions already hold for 
(Nedela and Škoviera 2022, Proposition 4.1).
