Necessary and Sufficient Conditions for Hamiltonian Based on Linear Diophantine Equation Systems with Cycle Vector

Two necessary and sufficient conditions are presented for Hamiltonian cycle problem in simple undirected graph using linear Diophantine equation systems with cycle vector. The first one is based on the incidence matrix and the second one is based on edge-adjacency matrix. It is proven that the solution set of the cycle vector correspond to the edges of Hamiltonian cycle in a given graph. Based on these result conditions, two necessary conditions for the Hamiltonian graph are given by determining the rank of the matrix.