Abstract
The saturation number of a graph F, written
$$\text{ sat }(n,F)$$
sat
(
n
,
F
)
, is the minimum number of edges in an n-vertex F-saturated graph. One of the earliest results on saturation numbers is due to Erdős et al. who determined
$$\text{ sat }(n,K_r)$$
sat
(
n
,
K
r
)
for all
$$r \ge 3$$
r
≥
3
. Since then, saturation numbers of various graphs and hypergraphs have been studied. Motivated by Alon and Shikhelman’s generalized Turán function, Kritschgau et al. defined
$$\text{ sat }(n,H,F)$$
sat
(
n
,
H
,
F
)
to be the minimum number of copies of H in an n-vertex F-saturated graph. They proved, among other things, that
$$\text{ sat }(n,C_3,C_{2k}) = 0$$
sat
(
n
,
C
3
,
C
2
k
)
=
0
for all
$$k \ge 3$$
k
≥
3
and
$$n \ge 2k +2$$
n
≥
2
k
+
2
. We extend this result to all odd cycles by proving that for any odd integer
$$r \ge 5$$
r
≥
5
,
$$\text{ sat }(n , C_r ,C_{2k} ) = 0$$
sat
(
n
,
C
r
,
C
2
k
)
=
0
for all
$$2k \ge r+5$$
2
k
≥
r
+
5
and
$$n \ge 2kr$$
n
≥
2
k
r
.