In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is not Hamiltonian. The result has been widely used to prove that certain planar graphs constructed to have additional properties are not Hamiltonian; for instance it can prove non-Hamiltonicity of some counterexamples to Tait's conjecture that cubic polyhedral graphs are Hamiltonian. Grinberg's theorem is named after Latvian mathematician Emanuel Grinberg, who proved it in 1968. Formulation A planar graph is a graph that can be drawn without crossings in the Euclidean plane. If the points belonging to vertices and edges are removed from the plane, the connected components of the remaining points form polygons, called faces, including an unbounded face extending to infinity. A face is a if its boundary is formed by a cycle of and of the graph drawing. A Hamiltonian cycle in a graph is a cycle that passes through each vertex exactly once. Let be a finite planar graph with a Hamiltonian with a fixed planar drawing. By the Jordan curve theorem, separates the plane into the subset inside and the subset outside every face belongs to one of these two subsets. Denote by and the number of faces of the drawing that are inside and outside respectively. Then Grinberg's theorem states that The proof is an easy consequence of As a corollary of this theorem, if an embedded planar graph has only one face whose number of sides is not and the remaining faces all have numbers of sides that are then the graph is not Hamiltonian. To see this, consider a sum of the form given in the statement of the theorem, for an arbitrary partition of the faces into two subsets, counted by numbers Each face whose number of sides is contributes a multiple of three to the sum, because of the factor in the term to which it contributes, while the one remaining face does not. Therefore, the sum is not a multiple of three, and in particular is not zero. Since there is no way of partitioning the faces into two subsets that produce a sum obeying Grinberg's theorem, there can be no Hamiltonian For instance, for the graph in the figure, all the bounded faces have sides, but the unbounded face has 9 sides, so