# On non-Hamiltonian cycle sets of satisfying Grinberg's Equation

In [1] we used a cycle basis of the cycle space to represent a simple connected graph G. If G is Hamiltonian, then there exists a set of cycles to be a solution of Grinberg's Equation. The result of the symmetric difference of these cycles is a Hamilton cycle of G. It is clear that a graph is not Hamiltonian if its Equation has no solutions. Conversely, not all graphs satisfying the Equation are Hamiltonian. In this paper, we reexamine the cycle combinations of Equality (3.3) in [1] and find a distinctive cycle set underlying the non-Hamiltonian combinations satisfying Grinberg's Equation. We detail the characteristics of this cycle set and the existing condition.

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