  Let G (V, E) be a simple graph with vertex set V and edge set E. A
generalized cycle is a subgraph such that any vertex degree is even. A simple
cycle (briefly in a cycle) is a connected subgraph such that every vertex has
degree 2. A basis of the cycle space is called a cycle basis of G (V, E). A
cycle basis where the sum of the weights of the cycles is minimal is called a
minimum cycle basis of G. Grinberg theorem is a necessary condition to have a
Hamilton cycle in planar graphs. In this paper, we use the cycles of a cycle
basis to replace the faces and obtain an equality of inner faces in Grinberg
theorem, called Grinberg equation. We explain why Grinberg theorem can only be
a necessary condition of Hamilton graphs and apply the theorem, to be a
necessary and sufficient condition, to simple graphs.

Discrete Mathematics