  In this paper we define a construct called a time-graph. A complete
time-graph of order n is the cartesian product of a complete graph with n
vertices and a linear graph with n vertices. A time-graph of order n is given
by a subset of the set of edges E(n) of such a graph. The notion of a
hamiltonian time-graph is defined in a natural way and we define the
Hamiltonian time-graph problem (HAMTG) as : Given a time-graph is it
hamiltonian ? We show that the Hamiltonian path problem (HAMP) can be
transformed to HAMTG in polynomial time. We then define certain vector spaces
of functions from E(n) and E(n)xE(n) to B = {0,1}, the field of two elements
and derive certain properties of these spaces. We give two conjectures about
these spaces and prove that if any one of these conjectures is true, we get a
polynomial time algorithm for the Hamiltonian path problem. Since the
Hamiltonian path problem is NP-complete we obtain the proof of P = NP provided
any one of the two conjectures is true.
