I was given this problem as one of my assignments in my class. I had tried a lot of math approaches including the mathematics induction and searched the internet but couldn't find such simple and accurate solution until one of my friends came up with it. Cheers to you buddy! Enjoy the brilliance of simplicity below.
Let
the cycle in d-dimensional hypercube has n0, n1,..,nk nodes. Processor labels
will change by only one bit to the adjacent nodes. As we go from node ni to
ni+1 there will be only one bit change in the processor label. Since, k is in
the order of 2k
the number of parity changes in bit must be even. Hence, there cannot exist an
odd number of cycles in a d-dimensional hypercube.
*Few Definitions:
Cycle: A cycle in a graph is defined as a path
originating and terminating at the same node.
The length of a cycle: The
number of edges in the cycle
