I hope this would improve its readability. I am a member of the organizing comitte of the first international mathematics competition for high schools. Now I will be glad if you send the solution of your following updated problems. And find the minimum number of bits required . G is a regular 2 - 2 graph 2.
G is connected.
It follows from prop. 1,2 that this process will succeed. We have divided G into vertex disjoint cycles. If there is only one cycle, we found a Hamilton cycle. We have joined two cycles together, and in this manner reduced the number of cycles.