Theorem (in any field of Characteristic 2):

In Graph Theory a number of results were devoted to studying the computational complexity of the number modulo 2 of a graph’s edge set decompositions of various kinds, first of all including its Hamiltonian decompositions, as well as the number modulo 2 of, say, Hamiltonian cycles/paths etc. While the problems of finding a Hamiltonian decomposition and Hamiltonian cycle are NP-complete, counting these objects modulo 2 in polynomial 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 even number of Hamiltonian decompositions in a 4-regular graph and an even number of such decompositions where two given edges e and g belong to different cycles (Thomason, 1978), as well as an even number of Hamiltonian cycles passing through any given edge in a regular odd-degreed graph (Smith’s theorem). The present article introduces a new algebraic technique which generalizes the notion of counting modulo 2 via applying fields of Characteristic 2 and determinants and, for instance, allows to receive a polynomial-time formula for the number modulo 2 of a 4-regular bipartite graph’s Hamiltonian decompositions such that a given edge and a given path of length 2 belong to different Hamiltonian cycles – hence refining/extending (in a computational sense) Thomason’s result for bipartite graphs. This technique also provides a polynomial-time calculation of the number modulo 2 of a graph’s edge set decompositions into simple cycles each containing at least one element of a given set of its edges what is a similar kind of extension of Thomason’s theorem as well. Additionally, it gives a polynomial-time algorithm for the number modulo 2 of a graph’s edge set decompositions into simple paths such that each vertex is an end of exactly one path, as well as the number modulo 2 of such decompositions into odd simple paths and into even simple paths.