HAMILTONIAN AND HYPOHAMILTONIAN OF GENERALIZED PETERSEN GRAPH GPn,6

Abstract

This article discusses the Hamiltonian and Hypohamiltonian properties of Generalized Petersen Graphs GP_n,6. A Hamiltonian graph is a graph that has a Hamiltonian cycle. So, a graph is called to be Hamiltonian and has a cycle that passes through each vertex exactly once. A Hypohamiltonian graph is if it is not a Hamiltonian graph, but if one vertex is removed it will be Hamiltonian. The Petersen graph is a cubic graph with ten vertices and fifteen edges and each vertex is of degree three. The generalized Petersen graph is denoted GP_n,k, for positive numbers n and k with 2 ≤ 2k < ????. The Petersen graph is not a Hamiltonian graph, but is Hypohamiltonian. In the Generalized Petersen graph for GP_n,6  for n ≡ 1(mod 13), n ≡ 3(mod 13), n ≡ 7(mod 13), n ≡ 9(mod 13) is a Hamiltonian, for n ≡ 0(mod 13) is a hypohamiltonian, and for n ≡ 2(mod 13), n ≡ 4(mod 13), n ≡ 5(mod 13), n ≡ 6(mod 13), n ≡ 8(mod 13) neither.