Solutions of Grinberg equation and removable cycles in a cycle basis
  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. 
Heping Jiang
cs.DM