The Balanced Hamiltonian Cycle on the Toroidal Mesh Graphs

The balanced Hamiltonian cycle problem is a quiet new topic of graph theorem. Given a graph G = (V, E), whose edge set can be partitioned into k dimensions, for positive integer k and a Hamiltonian cycle C on G. The set of all i-dimensional edge of C, which is a subset by E(C), is denoted as Ei(C). If ||Ei(C)| |Ej(C)|| 1 for 1 i < j k, C is called a balanced Hamiltonian cycle. In this paper, the proposed result shows that there exists a balanced Hamiltonian cycle for any Toroidal Mesh graph Tm, n if and only if m, n 3 and Toroidal Mesh graph nm 2 (mod 4), and how to find a balanced Hamiltonian cycle on Tm, n, for n, m 3 and mn 2 (mod 4). Keywords—Hamiltonian cycle; balanced; Cartesian product