  In this paper we provide some sufficient conditions for the existence of an
odd or even cycle that passing a given vertex or an edge in $2$-connected or
$2$-edge connected graphs. We provide some similar conditions for the existence
of an odd or even circuit that passing a given vertex or an edge in 2-edge
connected graphs. We show that if $G$ is a $2$-connected $k$-regular graph, $k
\geq 3$, then every edge of $G$ is contained in an even cycle. We also prove
that in a $2$-edge connected graph, if a vertex has odd degree, then there is
an even cycle containing this vertex.
