LOWER BOUND ON THE NUMBER OF HAMILTONIAN CYCLES OF GENERALIZED PETERSEN GRAPHS

In this paper, we investigate the number of Hamiltonian cycles of a generalized Petersen graph P(N, k) and prove that psi(P(N, 3) >= N center dot alpha(N) where psi(P(N, 3)) is the number of Hamiltonian cycles of P(N, 3) and alpha(N) satisfies that for any epsilon > 0, there exists a positive integer M such that when N > M, ((1-epsilon)(1-r(3))/6r(3)+5r(2)+3)(1/r)(N+2) < alpha N < ((1+epsilon)(1-r(3))/6r(3)+5r(2)+3)(1/r)(N+2), where 1/r = max {vertical bar 1/r (j)vertical bar j = 1, 2,..., 6} and each r(j) is a root of equation x(6) + x(5) + x(3) - 1 = 0, r approximate to 0.782. This shows that psi(P(N, 3) is exponential in N and also deduces that the number of 1-factors of P(N, 3) is exponential in N.