the number of copy of 6-cycles in petersen graph the number of copy of  6-cycles in petersen graph.I know that Petersen graph has ten copy of 6-cycles but I can't prove it.
 A: The following fact holds:

Let $G$ be the Petersen graph and $C$ be a 6-cycle in $G$, then the subgraph of $G$ induced by $V(G) - V(C)$ is a claw, vice versa.


In the figure above, orange is a 6-cycle and blue is a claw. There is a one-to-one correspondence between the 6-cycles and the claws. Thus we only need to count the # of claws.
Note that the degree of each vertex in Petersen graph is 3, implying that each vertex is the center of some claw. There are totally 10 vertices, thus 10 claws and thus 10 6-cycles.
A: Here are two 6-cycles in the Petersen graph:

By cyclic rotation, we generate the 10 6-cycles.  To prove these are the only $6$-cycles:
We must have precisely $2$ pink-to-blue edges.  To form a cycle, the number must be even, and there are not enough pink (or blue) vertices to have $0$, and we can't have $4$ because they cannot be connected to form a $6$-cycle.
So, deleting the pink-two-blue edges from a $6$-cycle gives a path on pink vertices and a path on blue vertices.  The path lengths must sum to $4$.


*

*The path lengths cannot be $0$ and $4$, since blue vertices are adjacent to only one pink vertex, and similarly for pink vertices.

*The path lengths cannot be $2$ and $2$, as the endpoints of the paths could not both be adjacent, due to the structure of the Petersen graph.

*So the path lengths must be $1$ and $3$, giving the already listed examples.

