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In Graph Theory a number of results were devoted to studying thecomputational complexity of the number modulo 2 of a graph's edge setdecompositions of various kinds, first of all including its Hamiltoniandecompositions, as well as the number modulo 2 of, say, Hamiltoniancycles/paths etc. While the problems of finding a Hamiltonian decomposition andHamiltonian cycle are NP-complete, counting these objects modulo 2 inpolynomial time is yet possible for certain types of regular undirected graphs.Some of the most known examples are the theorems about the existence of an evennumber of Hamiltonian decompositions in a 4-regular graph and an even number ofsuch decompositions where two given edges e and g belong to different cycles(Thomason, 1978), as well as an even number of Hamiltonian cycles passingthrough any given edge in a regular odd-degreed graph (Smith's theorem). Thepresent article introduces a new algebraic technique which generalizes thenotion of counting modulo 2 via applying fields of Characteristic 2 anddeterminants and, for instance, allows to receive a polynomial-time formula forthe number modulo 2 of a 4-regular bipartite graph's Hamiltonian decompositionssuch that a given edge and a given path of length 2 belong to differentHamiltonian cycles - hence refining/extending (in a computational sense)Thomason's result for bipartite graphs. This technique also provides apolynomial-time calculation of the number modulo 2 of a graph's edge setdecompositions into simple cycles each containing at least one element of agiven set of its edges what is a similar kind of extension of Thomason'stheorem as well.

A new algebraic technique for polynomial-time computing the number  modulo 2 of Hamiltonian decompositions and similar partitions of a graph's  edge set