I found the problem statement a little hard to understand, and I want to make some clarifications for anyone who is having difficulties.

1. We are looking for the maximum number of cycles such that each of these chosen cycles has at least one edge which is not included in any other chosen cycle.

2. The graph is not necessarily connected.

I originally thought that we had to choose the maximum number of cycles which have at least one edge not included in all other chosen cycles. Hope this can help people.