Hamilton Circuits in Graphs and Directed Graphs

Let h be an arbitrary n-cycle in S n . Assume h ' is another n-cycle in S n . Then if σ = (h) 1 − h ' , σ is an element of A n , the alternating group of S n . From elementary group theory, every element of S n can be written as a product of (not necessarily disjoint) 3-cycles. Thus, h ' = hσ 1σ 2 ...σ r where each σ i (i = 1,2,...,r) is a 3-cycle. Defining h i = hσ 1σ 2 ...σ i , we now stipulate that each h i is an n-cycle. Furthermore, let G m be a graph with vertex set V = {1,2,...,n} and edge set E = {e i │ i = 1,2,...,m}. Now suppose that h represents a pseudo-hamilton circuit in G m , say H, where H is a hamilton circuit in G " m = G m U { f i │ i = 1,2,...,s } with each f i an element of K n G m , where K n is the