TITLE: the number of copy of 6-cycles in petersen graph
QUESTION [5 upvotes]: 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.

REPLY [8 votes]: 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.