Number of Hamiltonian Cycles in Certain Bipartite Graphs of Size 4k+2

We will try to use the approach of labeled cycle cover sum to count the number of hamiltonian cycles in a special type of bipartite graphs, with 4k + 2 vertices which can also be considered as an alternative method to check hamiltonicity in such bipartite graphs. This graph is such that, there exists at least 1 vertex which is not a part of any cycle of size 4m + 2 where m k, in O(1.414n) time. The algorithm is inspired by Bjorklunds approach where instead of working on the field of GF(2k), we will be working on C.